Hi. A lot depends on your set up
But it's best never to do a "full" cage clean, but "partial" clean outs. What works well is if you have at least 5 to 6" depth of substrate and a litter tray with sand in (or some hamsters use their sand bath as a toilet). Empty the litter tray a couple of times a week. Then the rest of the cage will stay clean and dry - hopefully! You can spot clean occasionally if he pees anywhere else or if a lot of poops start taking over (the odd few isn't an issue - they even eat them sometimes which is normal). ie take out the odd handful of substrate and replace it with a clean handful - then mix it in a bit so it still smells familiar.
The idea is to keep familiarity for the hamster. Now with that depth of substrate and a litter tray the bottom of the cage and bottom half of substrate should stay clean and dry. For Syrians at least. For dwarf hamsters it can be a bit different as they can pee in a number of places so spot cleaning may work better than a litter tray - or a bit of both.
This way you can go a long time before needing to do a substrate change. When doing that though, it is best not to clean anything else in the cage at the same time - hence partial cleans - so something always has their scent and is familiar.
For a substrate change it's a good idea to keep back some of the clean old substrate (maybe the bottom section) and spread it on top of the new substrate - so they aren't too freaked about all their scent disappearing.
When doing a substrate change you can then clean the cage base - but I usually find it is not actually dirty, wet or smelly if there's enough substrate in the cage. But you can find a fair bit of sand and an extremely large hoard for example, so at that time you could "prune" the hoard - ie take away the bottom half. It's always best to leave the rest of the hoard though, if it is just dry hamster mix or they can get a bit anxious.
But just wiping the cage base with a warm cloth is fine I find. Some people use soapy washing up water,but rinse well afterwards. There is no need to use pet disinfectant unless there has been illness or disease. But if you do use it, they can have a scent and scent can really affect hamsters (they have a strong sense of smell and a sensitive respiratory tract that can be affected) - so if you do use anything like that rinse it really well and air it a bit so any scent disappears. Easier and quicker just to use soapy water and rinse, so the hamster isn't out of the cage for a long time.
Even when doing the substrate change, I leave the nest and most of the hoard in tact. Try to take the house out with the nest intact inside it and put it back again - sliding something under the house before you remove it can help keep the nest in tact.
Assuming their nest and house is clean and dry you can just put it back again. If they have pee'd in the house and nest then you'll need to wipe that and remove most of the nest, but try to leave part of it - a dry bit - even if it is slightly whiffy. Then put a new pile of nesting material out in the cage (not in the house - they like to forage for it to rebuild the nest). Plain white toilet paper sheets torn into strips is best - safe to pouch or if swallowed.
But don't clean anything else at this stage. You can then clean the wheel as and when needed,in between - or a different week. And any other cage items a different week again. Toys and tunnels etc often don't need cleaning very often or don't get soiled.
It is our human nature to want to clean everything all in one go, but not good for the hamster who feels like everything familiar has gone, it doesn't smell like home any more, and they can often get so stressed they want to escape and start bar chewing or other stress behaviours.
Hope that helps! I discard hamster bedding and sand in the normal waste but it depends what your local area waste rules are. We only have normal waste or recycling (which is paper,plastic and tin). So anything else goes in normal waste if its not recycling. Hamster waste doesn't count as "toxic" so doesn't need to be disposed of in a special facility.
Also if the hoard is pee'd on and does need to be removed, try and keep back a bit that is dry and leave it where you found it, plus add new food to replace some of what you've removed (just a tablespoon of new food is ok - it doesn't have to be as big as the old hoard).
Their nest and hoard are their most precious things they don't like being messed with - so if they're clean and dry just leave them. Dry hamster mix lasts for really quite a long time without spoiling. I find they sometimes mix sand in it as well (to preserve it!). Not accidental as the sand has sometimes been carried some distance! Fresh food they tend to eat straight away rather than hoarding it so you're unlikely to find rotten veg etc. The key there is only give them a tiny piece each day.
As a guideline - for our Syrians (one in an 80cm cage, one in 100cm cage) I only do a substrate change every 2 to 3 months - because they both use a litter tray. In between I do a bit of spot cleaning now and then - but not that often as they use a litter tray. It's usually more like adding the odd handful of new substrate after it has got a bit compressed and got lower - or after they have taken some substrate to add to their nest so it's got lower. I empty their litter trays every few days.
When we had a robo he peed all over the place - but not in his house or nest so I left those alone. For him I found it easier to just change half the substrate at a time - eg the left half one week, then the right half about 3 weeks later because I never knew where he pee'd and it was such tiny amounts it seemed to evaporate!