The number one most effective thing you can do to winter-proof your ride (and keep your backside and face clear of manky sprays of mud and god knows what else) is to pop on some mud guards. Handily they are also one of the easiest pieces of kit to attach too.
Choosing mudguards for a mountain bike is slightly trickier than for road and commuter bikes, because you’ve got front and rear suspension. No full mudguards for you, sadly. Although you don’t have to worry about tyre width, for some mudguards you’ll need to check whether they fit the wheel size you have; 26”, 27.5” or 29”.
For the back, there are several options. Clamp on mudguards like the Crudcatcher Raceguard and the attach to your seatpost and sit out behind you to catch the spray from your rear wheel. However, if you have a dropper seatpost, you may not be able to fit one as they’ll get in the way of the mechanism.
On the front, there are a number of things you can do that, while they won’t stop you getting wet feet, will at least stop a rain of spray or clods of mud flying up and hitting you in the face. The Rapid Racer Neoguard is an easy to attach neoprene mudguard that pop onto your forks. Mucky Nutz do a range of mudguards that price up at only £8.99. If you want something more robust that will give you protection from rocks as well as mud, the pricier but oh-so-cool looking Rock Guardz is a carbon-fiber front mudguard.
Most of these are simple to fit, but do be aware that if you have a carbon framed bicycle you should avoid using cable ties to attach these directly to the frame as they can cause damage.
If you do want a more whole-wheel coverage, then the SKS Bluemels MTB Guards offer this. However, the size range only covers tyres 1.0 to 2.1 wide, and you may find they get clogged up with mud quickly.