I've asked all of my trail running friends and the general consensus is to simply hose them off in the sink (some even just wear them in the shower...). I'm concerned that leaving soaking wet shoes just near the radiator will impact the longevity of the shoe. It also means I then can't wear them for a couple of days because they are wet.
What is an actual, proper care routine for my trail-running shoes? They are Brooks Cascades if that matters.
I hose mine or throw them in the washer (front load). Remove inserts first. I don't see why they would take a couple days to dry. They should have a good bit of mesh and dry pretty quick. Even if they are damp they will dry fast once you put them on. I sometimes set mine outside in the sun or flip them upside down on a register. It sounds like you don't have forced air if you have radiators so the latter wouldn't work for you but you could put them on the radiator perhaps. Even though I doubt this would have an impact on longevity unless perhaps they have real leather, which isn't common, the soles of trail runners will wear out long before any such issue occurs.
I backpack rather than run and am constantly walking through streams all day long. It's no big deal if your shoes are wet. They may not be fun to put on after below freezing nights but will warm up quickly once you're moving. FWIW, I wear Inov-8 Roclite shoes.
You may want to wash them because they stink too much for where you need to keep them. If it's just about mud, just brush them. Mine tend to get smellier if I've run through puddles or heavy rain, and then they're already wet so a wash is no problem.
With mine, or my bike shoes, I take the laces out and remove the footbeds, and first rinse/brush off the visible dirt. Then I use sports wash (marginally better than normal detergent for stink removal) in a sink full of warm water. A good rub of all the fabric bits before and after a bit of soaking is all that's needed. Rinsing afterwards involves running water through them, soaking in clean water, then running more water through them.
Then either put them outside in the sun and wind or stuff with newspaper and put under a radiator. Change the newspaper a couple of times, once after not long and once after a few hours. After another few hours remove the newspaper completely. This will take out a lot of water. You can also use absorbent rags or paper towels.
Drying on a radiator won't destroy them outright but isn't a good idea as various components will soften. It's only worth doing if you really need them again quickly - and allow to cool before wearing in case any glue is softened.
These are trail running shoes, not dress shoes that you will wear to the opera or to a job interview with a stuffy firm. They will get dirty on the next run, or the run after that, so there is little need to get them pristine after every run.
Therefore, I suggest waiting until the mud has dried, and then brushing the dried mud off. For seams, and the treads on the soles, use a pair of tweezers, or a small screwdriver, followed by brushing with an old toothbrush. If they still don't look good enough for your taste, take a damp cloth of some roughish material (i.e., a small old towel as opposed to a piece of an old pillowcase) and swab the shoes. This will leave them slightly damp, but far from soaking wet.
If there is no mud, but only dry dirt and other clean organic matter, do the same, but the step of waiting until mud has dried is, of course, not necessary. If you have stepped in feces, then you need to get the worst off with a paper towel, then carefully aim a hose with the "beam" at its narrowest at the spot(s), wipe them as dry as you can, followed by brushing, if necessary.
I normally don't wash them. Let them dry. Brush hard with a bristle floor scrubbing brush.
I don't worry about them being wet either. They are synthetic generally and won't rot. I backpack in trail shoes. They are normally wet all day. At night I change to crocs, but my shoes are still wet the next morning. Put on wet socks and wet shoes. I know I will be crossing a creek again within 20 minutes walking.
If your shoes have leather in them, you can use saddle soap on the the leather. This both cleans it, and replaces oils in the leather keeping it flexible.
If your shoes tend to get stiff as they dry, put them on when slightly damp, and wear them until dry. Feet are highly water resistant.
Most things have been said. Anyway, after cleaning, it's not a good idea to put them under or near the central heating.
Although it's not bad for a shoe to wear them while wet, it can cause blisters.
This means I only clean them when it's really needed and the easiest is:
