State old age pension is taxable only in France under article 18 of the Convention and private and corporate pensions are again taxable only in France under article 18.
“Government” pensions for service to the British government (eg. from the emergency services, military, government and diplomatic service) are taxed in the UK, but are taken into account in France in the income tax calculations in order to effectively increase the tax rates at which other income that is taxed in France (such as pensions other than “government” ones) are taxed at.
So yes, you can claim the tax overpaid, whether from the UK on the state pension and private and corporate pensions, or from France if they have taxed (and not just “taken into account”) any government pensions.
If you have not done so, you should notify the UK that you are French tax residents now so you are not paying UK income tax on income that should only be taxed in France. This is usually done on Form France-Individual, which you can find at gov.uk.
UPDATE: As of this year’s declaration (for 2018 income), non-residents and others who are attached to another EU/EEA country or Switzerland’s social security system for their healthcare (this includes British state pensioners in France who have healthcare paid for by the UK and have an S1 form) are being asked to select box 8SH or 8SI in the 2042C complementary section during their tax declaration to draw attention to this.
It means UK state pensioners in France are entitled to an exemption from French social charges on their UK pension incomes.
This question was answered by Olaf Muscat Baron who is a Fellow of the Chartered Association of Accountants UK, a French expert comptable and an International tax advisor.
He is the principal accountant of Fiscaly, an accountancy firm based in the Dordogne.
See www.fiscaly.fr or call 09 81 09 00 15
Email your tax questions to firstname.lastname@example.org