This is one of those issues that falls between the various legislative powers.
Traffic calming, as has been stated, must affect the movement of vehicular or other traffic and improve safety as a result. Traffic calming measures are defined in the Highways (Traffic Calming) Regulations 1999. In my opinion, a new footway is unlikely to be considered as traffic calming, unless the road becomes narrower as a result of installing it, so I'd be reluctant to rely on S274A for this. There is S43, which embodies the power to maintain footpaths, but not to install or improve them.
As is so often the case, we find ourselves reverting to our old friend S137, although expenditure is limited and highway works are frighteningly costly, so this is not always a reliable fallback. If all else fails, there's the catch-all S111.