I am living in London, but moving to a suburb where my commute would take me through a park. Being rainy England, the track can get quite muddy and my current tyres don't seem to be suitable. What features should I look for in a tire for this purpose? The bike in question is am aluminum Trek Checkpoint. I currently have 32mm Continental Gator Hardshell tires, which are slick tires with puncture protection.
Generally, wide tires have lower rolling resistance than narrow ones, and they will be more comfortable due to cushioning. Hence, there's an argument you could go all the way up to a 45mm tire, the maximum I think your bike will clear.
For performance-oriented road riders, aerodynamic considerations might come into play, but these are probably irrelevant for you.
If you’re running tubeless for the flat protection benefits, that may be an additional argument for a wider tire. We know that tubeless tires don’t work as well on performance road bikes as they do on mountain or gravel bikes. This is likely because of high pressure and relatively small volume. A punctured 25-28mm road tire at 70-90 PSI deflates very fast. A 40mm gravel tire at 30 PSI deflates more slowly. I have a hunch that if you want to run tubeless, wider is better, all else equal.
If you want to run mudguards (aka fenders), you may need to limit the size of the tires you fit. It's best to maintain adequate clearance between the fender and tire. If there is too little clearance, you can get a branch sucked between the tire and the fender, and this can break the fender. I don't believe there are any other practical tradeoffs in going to a wider tire. You'll increase your coefficient of aerodynamic drag, but this is probably of little consequence here. I didn't see if Trek's stated tire clearance is 45mm with fenders or without; I suspect the latter. If so, then I suspect 38-40mm tires will fit and still provide a very nice ride.