I am in the middle of this process, which is almost exciting - not!!
First you must have an English Will to cover your assets and savings etc in the UK, else if you die whilst resident in France, you will in effect be intestate. Best way is to ensure that everything is jointly owned by you and partner.
You must then find a local Notaire who should enable you to draw up a French Will to deal with your French estate, and which can include something which says you want English Inheritance law to apply.
I suggest, that you also draw up a detailed list of assets, bank accounts, savings and investments, list of any organisations that make a claim on your estate such as credit card suppliers, org's in France that have prelevements on your French bank account, and etc.
My advice would also be to get an accountant or maybe two one English if you have a complex financial situation in the UK and one French to deal with the financial aspects of your estate in France especially inheritance tax and ensure that all three/ four, English solicitor, French Notaire and accountant(s), all have the same set of documents.
Don't forget to make sure that your beneficiaries are aware that they are indeed beneficiaries and where the documents can be found (i.e. with the solicitor, notaire and accountant(s).
Finally all these documents need updating every year.
My best advice, don't die in France, though I accept that may be difficult to arrange. Not because it is France, but because you are overseas somewhere and each countries procedures are different and some time conflicting and which takes precedence.
And last but not least, I am describing as best I can the administrative process based on my personal experience but not the legal process for which you need professional advice. I am sure that others on SFN cab add their own perspectives.
What we need is a European Federation andn life would be so easy, but then that is another matter.!!!
Enjoy
Peter