Best way to rehome a cat?

Advice needed!
I live in rural France, have my own 2 dogs/2 cats. family returning to Uk wanted a cat fostered. I am the fosterer but they do not want the cat. She is adorable, small for her age, very pretty and friendly. She had kittens, they kept 2 and they say her nature changed. She does not like to be picked up very much but otherwise seems friendly.
She´s been spayed, innoculated, chipped and is passported and can enter the UK from Sept/Oct time.

I usually return to the UK during the winter (with all of mine in tow!!) Would I be able to find her a kind forever home in the UK? I worry that here in France she may get "passed on" or be kept as a barn cat and she doesn´t seem to be tough enough for that life.
Any ideas gratefully received.