Morning Firebird,
Your case involves what is called “triangulation” – you can google it. The triangulation simplification within EU allows an EU VAT registered intermediate supplier to buy and sell goods from/to other VAT registered EU businesses who do not belong in their own Member State without having to VAT register in one of the Member States in which the supply of goods takes place.
However, post Brexit, as the UK is no longer in the EU, a UK VAT number cannot be used to allow triangulation. Within EU Member States there is no VAT registration threshold for Non-Established Taxable persons, i.e. businesses with no fixed or business establishment in the Member state where a supply is deemed to take place. Therefore, the UK business will be required to VAT register in either France in your case on the basis of the onward taxable supply of goods. Local advice will need to be sought to assist with the French VAT registration.
Once the UK business is VAT registered in any EU Member state, you will be able to use that EU VAT registration number to facilitate future triangulation supplies , where their VAT registered suppliers/customers belong in 2 separate Member States.
The below link would be of help: https://www.inspireaccountants.co.uk/inspire-blog/new-vat-rules-after-brexit-triangulation-of-goods-part-three/