Janice, thank you for your prompt reply.
You relisted the question as Incomplete Answer before I had even provided an answer.
As UK residents, your worldwide income is chargeable to UK income tax.
Having said that, you can claim foreign tax credit relief against foreign tax paid in France on profits from gites business. Profit from this business would be chargeable to French income tax even if you are non resident in France for local taxatiion purposes.
So, if you were to leave the UK and settle in France, profits from UK property income would be chargeable to UK income tax. As UK national you would be entitled to your UK personal allowance against this rental income before tax is calculated.
I am not an expert on French taxation but I presume they have similar system to the UK. If rental income from the Uk were to be taxed in France then you would be able to claim foreign tax credit relief in France against it under double taxation agreement with France.
Furthermore, you may wish to read information on "The Non-resident Landlord Scheme” here
I hope this is helpful and answers your question.
If you have any other questions, please ask me before you rate my service – I’ll be happy to respond.