Hi America
If you ever wish to relocate FROM France at some point I'd be very careful about this.
At the moment your children's 'country of habitual residence' is one of the UK ones. English (and presumably scottish etc) law states that for a child to be taken out of the country the permission of all those have parental responsibility must be given. I am assuming for these purposes that your ex has PR and would be pleased to have the children in France, if just for convenience if nothing else.
'habitual residence' is not a term which is strictly defined in English law - its more of a 'which place is central to the children's day to day and ongoing life' term under which a court would look at many factors - eg, where they live most of the time, where they go to school etc.
It is likely that if you relocate to France the children's country of habitual residence will become France. France and the uk are all signatories to the hague convention. This means you may not relocate the children back to the uk or any other Hague convention signatory country without their fathers permission. He can withhold this permission for any reason he sees fit, even if he never sees them. It is about where the childrens place of habitual residence is.
If he doesn't give his permission for you to relocate the children from France you will have to apply to the French courts for 'leave to remove' which they may or may not grant, depending on a whole host of reasons, some of which may be particular to the French legal system.
If you relocate the children back to the uk without his permission he is able to bring an application under the Hague convention against you for child abduction. If he did so you will have to stand trial and the English courts will most likely order the children to be returned to France (from where you could apply for 'leave to remove' but with a bit of a blot on your character). Other Hague signatory countries (brazil, Sweden, and a few others) although signed up to Hague have a bad track record of complying with it. The uk is not one of them.
As to divorce, you need to look into which jurisdiction (England or France) would give you the best settlement. London is known as the divorce capital of the world. It is known for being quite archaic in some respects in terms of making gender differences and tending to be more sympathetic to mothers than other legal systems. I know nothing about the French judicial system - perhaps if you reported in 'Legal' someone might be able to help you.
What I do know is that the uk and France are both party to the Brussells II convention and as a rough rule of thumb, if husband and wife both file for divorce in different EU countries, the jurisdiction of the party who filed first will take precedence.
If you want to file divorce proceedings in England, you need to do so before you leave. In order to file divorce proceedings in the uk you need to be 'resident' here. Again, a term not strictly defined in English law, but will be difficult to prove if you have relocated. If you can't claim residence you could also try filing under 'domicile', an even more woolly legal term under which the court will look at all manner of things such as where you live, work, have bank accounts, are registered for tax, have mobile phone contracts etc etc. there will be a lot of case law on it and you would be taking a real chance if you didn't file before leaving the uk.
Get proper legal advice before making any decisions. www.reunite.org.uk will have a list of lawyers sufficiently experienced in international family matters.
Good luck my dear. I wish you the very best for you and your children.