The recent High Court judgement on the case in question can be found here, where it has decided that the fact finding judgement decided upon in December 2019 should be made public. In particular, it should be noted that this is not a criminal trial, but proceedings in family court, which has led to a so-called Fact Finding Judgement (FFJ). These take place to establish as fact allegations, which may or may not be criminal in nature, so that they can be used in a family court case. The family court case in question is Sheikh Mohammed suing his wife for custody of their children - not the other way around:
On 14 May 2019 the father commenced proceedings in England and Wales
under the inherent jurisdiction of the High Court seeking orders for
the children to be returned to the Emirate of Dubai.
As he is a chief of state, isn't he outside of the jurisdiction of a foreign tribunal due to diplomatic immunity?
This is not a foreign tribunal, but a fact finding hearing ordered as part of a family court case. In addition, as the Sheikh initiated the proceedings that have led to this fact finding hearing in the first place, he has put himself under the jurisdiction of the court by default.
You are correct that in general, under international law, heads of governments and states are granted immunity from judicial proceedings, however, as stated in the linked judgement, the Sheikh has signed a waiver of immunity in this case.
What legal basis does the princess' action have?
The princess has made certain allegations that are the subject of the recent judgement. The legal basis that this has is that her allegations have been acknowledged to be factual by the High Court, allowing them to be presented as fact by the Princess in the Sheikh's proceedings against her.
What power does the court in the UK have over the enforcement of the decision in this case? Can they only enforce parts of it?
With regard to the FFJ, the court only has the power to establish the allegations as fact. There is no enforcement to be made. Whether this fact finding judgement will result in criminal proceedings remains to be seen - Cambridgeshire police reviewed their investigation of this case in 2017 and confirmed that it was no longer active due to lack of evidence, but reserved the right to reopen the case if new information came to light. The Sheikh is unlikely to waive his right to immunity in this case, however.
Now that the allegations have been established, they can be used in the family court case initiated by Sheikh Mohammed, again, due to the immunity waiver he has signed. This makes it rather unlikely that the Sheikh will succeed in his private legal action against the Princess.
Why are UK taxpayers funding this trial?
Funding of an independent judiciary is key to upholding a functioning democracy. In addition, the two parties in the case are private individuals, so they are responsible for their lawyers' fees.