Planned works on the French side include new construction east of Lyon to replace the current slow sections of classic line, in addition to the new tunnel.
At Susa in Piedmont is a giant "NO TAV" sign above the town on the hillside - an indication of the local opposition.
en.wikipedia.org
en.wikipedia.org
The No TAV movement is well supported (I bumped into some of them when in Turin - they have/had a presence in the same building as an organisation I have links with). I assume the base tunnel will happen, which isn't just faster than the existing Frejus tunnel, but bypasses the first towns on each side of the Frejus route, so that'll speed things up a bit anyway (not that it's "High Speed" in the way that some new-build projects are promoted - just 140mph I believe - but it's massively straighter of course!). And I gather there's serious work on the French side. But I wouldn't bet on the tunnels on the Italian side from the base tunnel mouth onwards towards Turin happening any time soon. The local opposition, supported by national organisations, seems unlike the responses to other higher-speed rail enhancements in Italy; I presume the rest of the project would still happen, and be seen as worth doing, without the work on the Italian side. And my money's on that outcome for the foreseeable future.
In any case, even with the existing route, I've found London-Turin (and vice versa) within a day is perfectly doable, and I'm often happy to sacrifice journey speed for a pleasanter view!
Also, I suspect a major argument for the works is improved capacity on the route, as much as for decreased journey times, so the new base tunnel will help with that anyway given the restrictions using the old Frejus tunnel.