Since I am far from being a specialist and have no personal experience with this, this is more a comment than an actual answer but it's way too long for the comment box. Until someone else posts something better, I thought it could be useful…
My understanding is that you are not necessarily a resident for a whole tax year. You can be a resident starting on a given date or up to a given date, with different consequences depending on the country and just like with the earlier question about moving to France, it would seem to make the problem disappear.
Importantly, the six month/most of the time test is only a way to define who is a resident and who is not but if you do happen to stay in a country for long enough, you can be deemed a resident from the day you moved to the country, not for the past or coming tax year and not from the day you went over the six month limit (it's certainly the case in France, but I would think other countries have similar rules).
To see how it works out in your case, you would first look at the UK's definition of residence and at the French definition of residence and determine whether you are a resident on any given date during the year.
If by any chance, the difference in tax years or any other rule make you a resident in both countries for (long) parts of the year, the treaty should kick in.
For the purposes of this Convention, the term “resident of a
Contracting State” means any person who, under the laws of that State, is
liable to tax therein […]
Where by reason of the provisions of paragraph 1 an individual is a
resident of both Contracting States, then his status shall be determined in
accordance with the following rules: […]
Nowhere does it say that it applies to a tax year or something like that. Based on this interpretation, if, on any given day during the year, you happen to be a resident in both countries, then the tie-breaker rules should apply. And if everything else fails, the same article provides that
if he is a national of both Contracting States or of neither of them,
the competent authorities of the States shall settle the question by
So it seems you could approach both tax offices with the question since, under the terms of the treaty, they should in any case ensure that you are not considered a resident of both countries for tax purposes.