The principle behind the Care Act is that the family home does not have to be sold to pay for residential care of somebody who lived in the property before they went into care.
The LA lend money secured against the property, the debt plus interest has to be repaid after the death of the care home resident, at this stage it may or may not be necessary to sell the property.
In the OP's case it is hard to see how the French property can be classed as the family home as the family appear to have voluntarily chosen to move out and live elsewhere.
if a property is in the UK the LA put a charge on the property registered at the Land Registry, this is the amount they are prepared to lend and can be considerably less than the current market value of the house.
With the complexity of French ownership,the Euro/GBP variations, and the uncertainty of the market at death of the resident, I doubt if the LA would agree to much of a deferred payment.
Unless the family intend to return to the property in the future it's probably best to just bite the bullet and sell.
The alternative is to find the cash for funding elsewhere as, given some form of joint ownership, and unless the property is a complete wreck, Mum has over the lower limit of assets.