I did it east to west around this time of year. on a cotic escapade with cheap 35mm tyres, went with bikepacking style luggage. it was absolutely fine. Bone dry for the most part. There are quite a lot of stretches along old roads so at times the cycling can be pretty quick
There are some really lovely bits, and as mentioned before some fighting through hedgerows and across fields with your gps happily telling you you are "cycling on the icknield way"
but do it. the beacon didn't seem all that far so i carried on to the ridgeway until eventually i ran out of gears in the chilterns and my knee started getting iffy. ended up at a place called radnage where i camped, turned round and cycled pretty much the same way home again.
are you camping? i can point you to a superb wild camping site if that's your type of thing?