Answer

By Country Walking magazine

Unfortunately, we don’t know of any software or mapping that shows stiles - primarily because many of them use data from the Ordnance Survey which, as you say, doesn’t include that info.

However, there are a few other ways to investigate potential routes without stiles. The satellite view on Google maps zooms in pretty close in some areas, or try a site called Geograph which collects photos from every UK grid square and may show what you need to know. Local National Park authorities or tourist information services should also be able to provide info about accessibility.

In addition, there are a number of sites dedicated to walks suitable for wheelchairs: Walks with Wheelchairs, Walking on Wheels (Scotland only) and the Fieldfare Trust.
Also worth a look is the Miles without Stiles initiative which has good routes in the Lake District and increasingly across the country. We don't know of a central site but a Google search returns a stack of suggestions.

Certain long-distance routes are also increasingly accessible and stile free. The South Downs Way (see pic) team has done a lot of work replacing stiles with gates and the southern section of the Shropshire Way we also know to be completely stile free.

Another potential avenue is a series of walks written for families with pushchairs, with a focus on no stiles or kissing gates, see Pushchair Walks and Walks with Buggies.

And finally, the routes in Country Walking magazine always specify whether there are stiles, or if you like the look of a particular area, ask on the forum here for expert local advice.