According to the Sustrans maps, there appears to be traffic-free cycle paths the entire way. Won't cut down on the miserable weather, but maybe will cut down on the puddles and mud?
It is highly unlikely to cut down on anything but your speed of travel.
'Traffic-free cycle paths' are not necessarily free of puddles and mud - far from it. Every single grass or earthen bridleway in England and Wales could be described as a traffic-free cycle path; many stretches of the TPT are shared with horseriders, and all of it with walkers. Neither horseriders nor walkers actually
prefer to travel long distances on foot on tarmac or concrete...
In addition, large stretches of the TPT pass through areas where there have long been what I will describe as access issues - in that off-road, often unlicensed motorbikes and the like vehicles make, or attempt to make, frequent incursions, causing damage, disruption and even danger. Hence there are concerted efforts to prevent access by other than legitimate users - as of course is proper.
Unfortunately this means that intrusive barriers - which are constructed so as to permit the legitimate user through, but not the illegal one - abound. These barriers provide various degrees of difficulty - sometimes at the level of impossible - for the loaded bike, the trike, the tandem, the trailer, the independent person with a disability etc etc to pass through. The horse barriers can make it difficult or impossible for a tiny pony, someone leading a second horse, one nervous in confined spaces etc.
If you go to the
TPT maps here and click on the symbols on the maps, you can see photos and read descriptions of the access barriers - and see how very frequent they are in many places! You will also see the conditions of the paths, they vary from tarmac to compacted stone to grass and earth.
Also bear in mind that neither the TPT nor the L-L towpath are intended, or engineered, for high speeds and in many places, especially where the surface is good and you might otherwise think you can get a move on, might - at least at certain times of the day - be busy with locals using them for their own leisure purposes.
I agree with
@Blue Hills above - plan yourself an on-road route using cycle.travel/map and make some diversions along the canal at places where there's something of interest, the towpath is good and there's a couple of decent pubs.