No, the RSPCA will NOT take the dogs away if they're living in a car - if they have shelter, food, water, exercise, their needs are met, then they can live in a car, as can the OP.
Jesus wept did some folk get out of bed determined to be as unfuckinghelpful as possible today?!
OP - sort the car as best you can for living in. Think about in particular, dealing with condensation and cold. There are lots of van/car living resources online, though a lot are very America-centric.
Once you are actually homeless, you'll have higher priority on the social housing lists with the local authority, so start bidding for the bedsits as soon as possible - the chances of you being evicted for having dogs that you aren't supposed to have is pretty much NIL and it would in any case, take them fucking MONTHS to do once you've signed the tenancy agreement.
That route, whilst not deal, should see you housed fairly quickly even if its not ideal, and give you space to save money to find somewhere better in the long term. Social housing being much cheaper to rent than private rental.
Alternatively you may decide you actually prefer living in a vehicle and could save for a van or possibly a boat (thats very dependent on how much money up front you have and how close useful waterways are to where you need to work of course but the actual running costs of live-aboard can be FAR lower than renting a house, even social housing!)
I wish you the best of luck, I will not be parted from my dogs unless that were to be in their best interests, which is highly unlikely.