Firstly, the tax year runs from 1 Jan to 31 December in France.
In the UK - You should only be taxed in the UK from the beginning of April (start of the new tax year in 2021) to the date that you leave the UK and you would contact HMRC to declare that you have left the UK.
In France you declare the income for the whole year, but file a form to effectively deduct the tax you paid in the UK.
The page here on the French government website details this, it is in French though -
You declare your earnings for tax in the May of 2022 for the year of 2021. This is the point at which you need to make sure you submit all of the correct paperwork tome sure you are credited for tax paid outside of France.
If you are concerned about this, there are bilingual accountancy firms that will submit all of the tax documentation for you, normally for the one off cost of around €250. You can find these firms by looking on google.
I hope that this answers your question?
Thank you for your enquiry today. I am happy to answer follow-up questions - please do get in touch with requests for extra information or further queries and I will do my best to help you.