It seems to be a combination of two factors.
Firstly, the time appears to have been chosen so that the restrictions will come into effect when there are no flights in the air. Yahoo has carried the best transcript of Transport Secretary Grant Shapps' BBC Breakfast interview on Friday morning that I can find:
Asked why it is the case that those who return to the UK from France
before 4am on Saturday will not have to quarantine for 14 days whilst
those returning after that time would have to do so, he said: “I think
the truth of this is, as everyone watching realises, there’s no
perfect way to deal with coronavirus.
“Unless you were going to have a sliding scale that sort of said if
you stay another 24 hours then you must quarantine for X amount of
time, another 36 hours for Y amount of time, you know, clearly there
has to be a cut-off somewhere.”
He added: “To be clear, the Joint Biosecurity centre have cleared our
approach to this.”
An estimated 160,000 holidaymakers are expected to be looking to
return to the UK from France following the government’s announcement.
Shapps said: “It’s a practical approach as well which has enabled all
fours parts of the United Kingdom – Scotland, Wales, Northern Ireland
and England – to implement the same time at 4am where there are no
flights in the air at least tomorrow.
“But, look, I accept your point, you can always argue one way or the
other. We have to make a decision on it and we have to do that based
on science and medicine, and that’s what we’ve done, we’ve taken the
advice and implemented on that basis.”
This reason does hold up to some surface-level scrutiny - one might ask "Why not Friday morning at 4:00?" - but according to FlightRadar24 at least one flight was in the air at that time - a flight from Paris Charles de Gaulle which took off at 3:53 am BST, landing in Newcastle at 5:43 am BST. On the other hand, a similar cursory check on Saturday morning doesn't appear to reveal any planes at all.
The second reason appears to have been to allow travellers to quickly return home before the isolation regulations kicked in. One might argue that this reasoning somewhat defeats the point of the restrictions, and it seems that the devolved administrations agreed.
The FT reports that UK ministers had originally wanted the restrictions to come into effect at 4:00 am on Sunday, in order to "give holidaymakers a greater chance of getting home before they would be forced to self-isolate for 14 days on their return" but that this plan was scuppered by delegations from Scotland & Wales who wanted a more immediate implementation.
As healthcare and the response to the pandemic are both devolved matters, the two administrations appear to have signalled that they would implement their restrictions earlier, and it seems that to reduce confusion the UK ministers agreed to an earlier implementation on Saturday.
This delay may also be due to lessons learned from the implementation of the two-week quarantine imposed on travellers from Spain from July 26th. In that case, the announcement was made on July 25th just after 6 pm BST, leaving a gap of less than six hours before the cut-off. After the deadline, there was then confusion as to whether the restrictions applied to islands such as Majorca, Ibiza and the Canary Islands, and the government faced criticism for the "shambolic" implementation, in response to which the Foreign Secretary said that the government "can't make apologies". A slightly longer gap before implementation is presumably an attempt to mitigate this response.