Firstly, are you sure it is the cats that are causing this. Very often doctors decide its the pets and don't do any specific tests. If they have not positively proved its the cats then I would advise trying to place them somewhere for a month as a first thing to try.
If it makes a difference then obviously you will need to home them permanently.
Most rescue I know of are full and you may have a hard time getting them into a rescue, but that doesn't mean a rescue can't help, for example we put cats on our website (no good for you as we are Coventry based).
As far as advertising them yourselves then start with family and friends, then vets and petshops. There are various free ad sites that can also be used, but you must vet the replies.
At 10 years old, it will be difficult to find new homes for them. If you contact local rescues they may know of someone who takes in older cats and would do so - especially if you were offering to support them.
I wish you well.