The Bath end is close to home for me, and I've ridden quite a bit of the rest.
The Kennet and Avon towpath isn't impossible on a road bike, but in winter it gets rather muddy especially west of Bradford on Avon. Combine that with a nice weekend day when it gets busy and you'll be in for a very frustrating ride.
Another tricky patch (mud and lumpy grass) is near Honeystreet (where there's a good cafe). That's just before Devizes coming from London; from Devizes it's decent gravel at least for a bit, as is the last stretch coming into Bath.
Through Reading there's some rather rough gravel, a little uncomfortable.
On the whole, I'd do it on my tourer with the tyres I normally run: 32mm slicks in summer, 35mm Marathon Mondial in winter, with a bit of grip. My road bike is a gravel bike - I'd use my road wheels (again 32 mm slicks) in summer, but my gravel wheels in winter (37mm grippy ones, can't remember what). Plenty of people I know would ride it on 25s, at least after dry weather.
It partly depends on the amount of time you have. I wouldn't route down long stretches on a road ride in winter, even with good lights, because it would be slow going. But if you're riding over a few days, that doesn't matter. Even in spring/summer, when I've been known to do Bristol to London and back in a day or slightly more, I avoided the towpath.