Hi Karen.
We only have one cat, but he's a long hair and truly provides more entertainment, love, and delight on a daily basis than anything else I can think of. The only guidance I was given was to completely avoid all contact with the litter (and for it to be changed often - when it was, I was in a totally separate room to avoid any airborne dust), and for the first 100 days or so, to wash and disinfect my hands immediately after petting the cat.
I think your decision to stay in a flat for the first month is a wise one, but I don't think you should worry about your house after that. Keep it as clean as you can, but you need to recognise how much better and relaxed you feel at HOME instead of somewhere rented. Sleeping in your own bed, with your own things, and familiar surroundings will be better for you than any slightly elevated risk of infection.
No one lives in a sterile bubble, but we live on a boat that's often damp, mouldy, dusty, full of cat hair, and midway through renovations (and will be for the next ten years). We of course tried to keep it as clean as possible when I was ill, but there's so many other sources of infection out there that I wouldn't write off the comforts of home just because it might not be straight out of Martha Stewart. And at the end of it all, my meningitis, EBV, liver infection, and estrogen depletion weren't caused or even exacerbated by anything lurking on the boat!!
I think hiring a cleaner for a few months would be a better use of your money than staying away from your home and pets for an extended period of time.
