Congratulations on your new baby OP!
This month: do you have any friends or family who could come and help as a one off?
Slightly longer term: get your 3yo to help (or "help") my DS has his own tiny sponge and duster and loves copying. We have a tidying up song. Treat of juice and biscuit when it's all done, or sticker for good tidying toys. Doesn't always go to plan but worth a try!
Sling for the baby - my oldest would go to sleep in it while I hoovered!! Every baby is different though, I don't think it was anything I did, it was just luck he liked that white noise
Cut right down on the number of products, it just complicates things. If you can find any multi surface products, they are your friend
Whenever you can, put stuff away don't dump it on a surface
I am the second least tidy person in the world (DH is worse!) but this keeps me out of actual health hazard territory
Good luck. You are basically superwoman with a three year old and newborn on your own. Be kind to yourself, you deserve it