A stuffy nose is common during pregnancy anyway.
I far prefer my Miele bagless to the Dyson I used to have. Vacuum every day, no excuses (giving birth just means somebody else has to do it instead). Make sure you do under the bed, behind furniture and the curtains/blinds.
Damp dust and swish a damp mop about every day - you'll be doing loads of cleaning anyhow once you're sterilising bottles and whatnot, it just adds a few minutes onto your daily stuff - it's easier to get into a routine before the baby is born than try to start afterwards. If you have carpets/rugs, get them cleaned.
Sheets on a 60 degree wash. Pillows, too (ideally replace and use an anti Dust Mite cover. Vacuum the mattresses weekly and if you can afford it, replace your duvet and have a Dust Mite protector inside the cover as well as one on the mattress.
Wipe the cats over with a damp flannel daily to reduce fluff and dander. They don't have to like it, but they'll probably get used to it, as it'll remind them of being cleaned my their Mum. Nuke all their bedding/blankets in a hot wash. Make sure washing is put away rather than have them use it as a better bed than the things actually meant for them.
Shower daily, even if it's just a few seconds under water to get anything out of your hair before you go to bed.
I also use boiled water with seasalt to flush out anything in my nose when it's particularly bad. Not exactly pleasant, but it really relieves symptoms.
A friend who has lung disease and allergies swears by his Dyson airfilter - it didn't do the job immediately, but after a week or so, he began feeling the difference. But he also keeps his flat perfectly clean - it would be pointless if he didn't.
If you can stretch to the cost of a Dyson, I'd try and afford a professional Deep Clean of your house first - they'll get dust and fluff from places you wouldn't have thought of/are unable to reach whilst pregnant. And I can't think of a better way to start than having a perfectly clean home.
There's no reason to get rid of the cats 
.