In my apartments, i kept cleaning supplies and bucket under the bathroom and kitchen sinks. Broom between fridge and wall. Sponge mop and, later, Swiffer, beside the fridge, too. Yes, inconvenient to not have bucket by sponge mop, but i often just wet the sponge at the kitchen or bath faucet and damp mopped. Vacuum in the front hall closet (along with wrapping paper and suitcases lol)

I always mop my kithen or bathroom floor sans bucket. I use the sink instead, rinsing the mop inder running water. i like having a clean mop for each section.

Look into a 2-in-1 moping and spray. My old apartment was small enough to clean the floors with a plastic ice buket I used instead of a larger bucket and towel. As long as your keep up then the a spray mop would be way better then the mop and bucket. I use a broom that has gripping holes that holds my dusting rags and mopping rags. Sounds like condencing is the key here.

I too like the idea of the spray mop. I have the Swiffer WetJet for wood and you could also use it with their fluffy dry pad instead of a broom (they do a better job than a broom and hold the dirt) or get the dedicated Swiffer for dry mopping. I haven’t used buckets for a very very long time because it feels to me I’d be using the same dirty water over and over plus they just take up too much room. I’ve also gotten down on my hands and knees with spray cleaner and paper towels since I feel they are cleaner than sponges and don’t need to be dried out like a sponge does. The other plus is that both methods take up hardly any storage space, so that would be my suggestion to solve the storage space problem, and save time too.

Another addition might be a robovac or a lightweight or hand-held vacuum just to keep up with things. We have a long-haired cat and get litter and cat hair everywhere. I run the robovac daily and also have a lightweight vacuum for quick pickup. I too don’t have ample closet space or a lot of time, so this works really well to keep the house very clean with very little effort.

So, here's what I did, for the archives. Once my new W/D arrived, there is just enough room to hang my broom and Vileda flat mop (that I use when damp mopping). I do like a larger, absorptive mop and bucket for bigger problems - I hung that mop behind my bathroom door (not ideal, but my door is almost always open, so I don't really see it), and right now the bucket is on top of the stacked dryer. They're not as easily accessible as I would like (I kind of have to curve my hand into a narrow gap to get them), but it does the job.

Right now my vacuum is still in a corner of the the front hall closet, and it might have to stay there :(

I hang up dusters etc behind bathroom door as when anyone opens it, it can’t be seen and less likely for visitors to see, bucket goes either under sink or behind door if space.