Frame challenge: I don't quite see what different jurisdictions has to do with this. The UK charged him for an alleged crime committed in the UK (obstruction of justice by failure to disclose something to UK police) regardless of what he didn't disclose, which was something [indirectly] about stuff that happened in France--and I say "indirectly" because the UK charge was apparently about not disclosing his phone password to UK police. He was not charged in the UK for failing to disclose something to the French police, AFAICT. So I don't see what "logic breaks down".
As far as jurisdiction goes, it's not different from the FBI charging a Russian with lying to the FBI about stuff that happened in Russia. Sometimes such questions can seem entirely legitimate (e.g. imagine "did you talk to someone in the FSB HQ in Moscow before coming here with a suitcase of cash"?) But of course, since lying [to the Feds] about any topic is indictable (like "what color was the condom you used yesterday in Moscow?"), it can be used for "gotcha" questions that might otherwise provide no indictable information if answered truthfully.
Whether the obstruction charge will stick in [a UK] court is another matter. Aside, there are various articles in the US press about obstruction of justice being a possibly charge there too for not disclosing a phone password, but they don't mention any concrete cases. (There is a later  article that says that the Supreme Court of New Jersey has decided in a 4-3 vote that not giving your phone password is obstruction. SCOTUS denied certiorari on that case in 2021.) OTOH guilty verdicts in rather similar (phone password) cases have been handed down in the UK. (The defendant in that case also lost a fist appeal and announced appealing to the UK Supreme Court, but I've not been able to find any further info on the aftermath.) I suspect that in general the law [and its case law wrt passwords] was not overturned because a UK police FAQ still says:
Can the police demand passwords for electronic devices ?
Yes. If a person is being examined, they can be required to provide any information requested, including passwords and PINs to any electronic devices. They will commit an offence if they wilfully fail to comply with this requirement.
The UK law applicable is less straightforward (than lying to the FBI) because the "schedule 7" investigation has to be plausibly linked to terrorism, which I suspect is more debatable when it comes to protests in France. I suspect the real challenge in court is probably going to be on that angle, on the case you've brought up.
From the press reporting you've linked, it's not even clear if the French claimed their investigation had anything to do with terrorism, or if that was decided UK-side. The BBC's reporting on that precise issue is also a bit ambiguous:
Officers said they were stopping him under Schedule 7 of the Terrorism Act 2000 - this gives the police wide powers to search people at border crossings to check if they are involved in terrorism.
The police do not need any grounds to stop and search people at borders under these powers.
So it may have been entirely a UK decision to throw the terrorism angle at him. Even his lawyers didn't say that France alleged terrorism involvement. In fact, they complained that he was given "no justification or explanation" for the password request:
"It was demanded that he give up his phone and pass codes to the officers, with no justification or explanation offered. This morning, Ernest was formally arrested and transferred to a police station, accused of obstruction because of his refusal to give up his pass codes."
Finally, I suppose a valid concern may be that a country with a law like the UK's may communicate information from the "answers forced under penalty of obstruction" to another country [like France], who'd then prosecute. That is a possibility, but it may also be possible to claim in defense that [in this case in France] that the information was indeed forced, so inadmissible. Whether that would carry legal water, depends on minutia of the specific country's legal system (as is evident in the New Jersey case, for instance.) OTOH, the ECHR has held a rather dim view of shenanigans involving documentation/evidence obtained that way:
Funke v France represents the European Court's earliest affirmation of right to silence protections under the Convention. French Customs authorities had sought various financial records in connection with an investigation into the violation of French Customs regulations. Funke was fined for failing to turn over the sought-after documents, but the Court concluded that the procedure involved impermissible
compulsion to produce self-incriminatory information in violation of the Convention.
"being unable or unwilling to procure [the documents] by some other means, [the government] attempted to compel the applicant himself to provide the evidence of offences he had allegedly committed. The special features of customs law . . .
cannot justify such an infringement of the right of anyone 'charged with a criminal offence', within the autonomous meaning of this expression in Article 6 , . . ., to
remain silent and not to contribute to incriminating himself.
Both Heaney v Ireland and Quinn v Ireland involved the application of an Irish law that criminalised the refusal of a suspect to answer police
questions concerning crimes under the Offences Against the State Act 1939. The Irish
State's attempt to justify using the threat of criminal punishment to compel potentially
self-incriminatory evidence on the grounds that "the information sought could be
essential for the investigation of serious and subversive crime" was rejected by the
Court. In its ruling the Court took the position that:
"the security and public order concerns relied on by the State cannot justify a provision which extinguishes the very essence of the applicants' rights to
silence and privilege against self-incrimination guaranteed by Article 6.1 of the Convention."
When information is sought by the State for purposes that are not strictly or solely, criminal, the applicable analysis is more complex. In some cases of this sort of criminal prosecution may lie just beneath the surface. Such was the case in Saunders
v United Kingdom, where the United Kingdom Department of Trade and Industry (DTI) undertook an investigation of an illegal stock support scheme implemented by Saunders in connection with a corporate takeover. Saunders complied with a directive
that he answer potentially self-incriminatory questions under threat of contempt, and his responses were used at his subsequent criminal prosecution for violating the Companies
Act of 1985. However, this was held by the European Court to violate Art.6(1) of the Convention.
As far ECHR is concerned, it would probably make no difference that information is compelled in the UK, but then used in a criminal trial in France, but that  paper doesn't discuss any such [cross-border] cases, perhaps because none had reached ECHR by then.
There is however this interesting case that adds some nuance to where the ECHR saw a limit to self-incrimination defense:
[In] Serves v France [...] a French officer and several comrades had been charged in connection with a homicide. However, because of a procedural
irregularity, the initial judicial investigation and all subsequent proceedings were
declared void. This was followed by a second investigation in which Serves, grior to
being recharged in connection with the homicide, was called to appear before a military investigating judge as a witness in the case on three separate occasions. Each time he
refused to take the oath and give evidence as a result of which he was subjected to three separate fines. Serves claimed this to be a breach of his Art.6.1 [...], but the European Court disagreed.
Although the Court recognized that Serves had a legitimate concern that some of
the evidence he might have provided could have been self-incriminatory, it concluded
that the proper response would have been for him "to have refused to answer any questions from the judge that were likely to steer him in that direction". Refusing
to take the oath and answer all questions, however, was not a permissible option. In
the Court's view, the French procedure involving fines for refusing to appear did not
represent impermissible compulsion to answer potentially incriminating questions, but
rather was only a means of ensuring truthful responses.
I don't know if any [phone] password compulsion cases have reached the ECHR. It's not exactly clear how that would be decided just based on precedent, because e.g.
Despite Funke, the status of efforts to compel the production of documents under
the Convention is unclear. In Saunders the European Court stated that right to Silence
protections are inapplicable to demands for: I
Material which may be obtained from the accused through the use of compulsory powers but which has an existence independent of the will of the suspect, inter alia, documents acquired pursuant to a warrant, breath, samples and bodily tissue for the purpose of DNA testing.
And the UK in particular seems to have taken an interpretation that would allow them to get such documents in such manner:
an Attorney General's Reference [(No.7 of 2000), Re, (2001) 2 Cr. App. R. 19] in the United Kingdom took the position that the decision European Court are inapplicable to all efforts to compel the production of documents, whether by subpoena or warrant, as long as they contain no compelled statements of
And in Nov 2022, French courts have held rather similarly wrt to phone passwords, specifically
The French Court of Cassation has ruled that people who are suspected or accused of a crime are obliged to reveal the passcode of their mobile phone to the investigative authorities. The Court found that a mobile phone passcode can be considered a “secret decryption agreement of a means of cryptology” (convention de déchiffrement d’un moyen de cryptologie). Refusing to hand over the passcode of a mobile phone is punishable by a fine of up to 270,000 EUR or three years’ imprisonment. This punishment is increased to a fine of 450,000 EUR or five years’ imprisonment where revealing a passcode and giving access to the content of the mobile device could have prevented a criminal offence or reduced its impact.
That article also notes laconically that
“Decryption orders” in France are currently under review before the European Court of Human Rights in the case of Minteh v. France.
So, as a kind of summary here:
Whatever that guy told the UK police under the compelled inquiry would be protected from use in a criminal trial (almost certainly anywhere) under ECHR jurisdiction.
Not surrendering his phone password to the UK authorities [which is what he was charged for] is perhaps not terribly relevant [for the cross-border aspect], since French laws currently allow direct (NJ-style) compulsion (and probably with greater penalties for non-compliance). Although the ECHR seems has yet to rule on such "decryption orders".