I did this the other way on a touring bike with 35mm tyres (Schwalbe Kojaks) last year. It's doable on a road bike, certainly.
Some substantial sections of it at the Reading end are not really surfaced: there is quite a lot of gravel, a fair number of rooty sections and one stretch is just a grass track through a meadow.
Also, where the Sustrans blazes divert you onto the road, I'd do as you're told. The stretches of towpath that are not National Cycle Network are narrow, rough and unsurfaced - you can end up doing miles along what is basically a rut through the nettles.
I don't want to put you off, it's a lovely route and I had a very good day doing it. But I got pretty tired hands and shoulders and made slow progress for long stretches at a time - it's about 90-100 miles and took me a solid 10 hours or so without lengthy halts. The weather was utterly gorgeous, which helped.