Better yet.....do whatever you have to do to buy a house.

Around here the number one reason people surrender a cat to
the shelter is the landlord won't allow it.

Shelters will check with the landlord before allowing an
adoption to go through.

Can't buy a house?  Some landlords will allow pets if you
agree to a larger security deposit.  

Talk to your landlord. If you live in something like a dormitory,
forget about it. It'll be found and out of there within a week. I guess
it would depend on whether the owner of your flat is a big or a small
business. If they're big and they have set rules, it would be hard to
convince them to make an exeption. If they're small, as in they only
run one or two small complexes, you might be able to make an agreement
of some kind. I had to beg my landlady to let me have a cat when I
moved in, and that includes making sure it was neutered and I can only
have one cat and I have to get rid of it if any other renters in other
flats complain about the smell or the cat being loud, etc. Like the
other guy said, you might be able to bribe them with a bigger security
deposit or paying so much extra every month, but I wouldn't pay more
than about 12-15 pounds a month. That's all they'd need to fix anything
your cat scratched up or peed on if you have a 12 month lease. If you
get them to allow you a cat, MAKE SURE YOU GET IT IN WRITING WITH YOUR
LANDLORDS SIGNATURE! Make them write it into the lease somewhere and
keep a copy of the new one yourself if they decide to be unfair and
want to throw out your cat without a reason you already discussed with
them. If none of that works, try your hardest to find flats that allow
pets. I don't know if that's just a general rule in London or what, but
there should be at least a few that allow small pets. I really wouldn't
try to get a cat without your landlords permission, because the only
thing harder than not having a cat is getting one, falling in love with
it, then having to give it up.