I’m not involved in importing/exporting, but my understanding is as follows:
Pre-Brexit, the treatment did depend on whether the customer was a VAT registered business. Where they were, no VAT was charged by the EU supplier and acquisition tax was declared by the UK customer on their VAT return. Where they weren’t, the EU supplier had to charge VAT in their country. Either way, no UK import VAT or duty applied to the movement of goods between the two EU member states.
Since the end of the Brexit transition period, I wouldn’t expect French suppliers to be charging French VAT on goods that they were exporting to the UK – regardless of whether the UK customer was VAT registered. I don’t have any references to French guidance, but if the transaction had been the other way around, HMRC’s notice 703 implies that a VAT registered UK supplier wouldn’t charge UK VAT on goods exported to a French customer, regardless of their VAT status.
What a UK customer is liable for is UK import VAT and duty – and the method of accounting for that can vary. VAT registered businesses may be able to arrange with HMRC to have it charged to a deferral account, or account for the VAT element via their VAT return. But an unregistered customer would have to pay the VAT at the border. In practice the VAT/duty may be paid by the carrier and the cost recharged. If shipping had been arranged by the French supplier, I suppose you might find that the cost of the UK import VAT and duty is indirectly recharged to you via them – but that wouldn’t make it French VAT.