Well, you don't need any trays (except maybe for use as a water-bath to place the developing-tank in, if the room temperature is extreme) and the chemicals stay inside the developing tank instead of being open in the air - so you won't have any smelly stuff filling the bathroom. Also, don't forget that these days you can even use smell-free chemcals too. I suggest going to the Ilford
website and browsing through their help documents to find the reduced water-useage archival-washing instructions, as that will save a lot of water and therebye reduce costs and waste.
The film has to get from the cassette, or roll, in to the tank without any light affecting it of course, so use a changing-bag. These are light-proof rectangular bags with a light-proof opening (to put all the bits and pieces inside the bag) and two light-proof armholes (to enable you to get your hands inside to load the tank). Once the film is in the tank, the chemical stages of the procedure are done in daylight, so you don't even need to be able to blackout a room.
If you are processing sheet film, then you can still use developing-tanks but they will be a different design than the ones used for the smaller 35mm or 120. It is not necessary to have deep-tanks, containing gallons of chemicals, sitting permanently in your bathroom.
The impact on the domestic environment is minimal, and the environmental impact with modern chemicals is also small. The normal chemicals are generally no more unpleasant than the bleaches and cleaning-products usually used around the home. If you produce umpteen gallons a day of photographic waste then the picture changes, but a pint a week falls well within regulations everywhere I have ever heard of.
EDIT: In the time I took to make a cup of tea and write the text, about half a dozen people have already posted !
