The problem is once you get north of the Midlands the timings become a lot more than "slightly slower". My wife and I travelled round France last summer and in terms of London - Nice the train couldn't compete on price let alone time!
London-Nice is doable for £62 if you get the cheapest advance tickets, but I'll concede that it's ten hours.
Part of the problem is that Nice isn't on the LGV network - LGV PACA would save about an hour - but the time you lose in Paris (or the rather poor connections at Lille for the trains that use the Paris bypass) is a bigger problem.
If you could teleport from Gare du Nord to Gare de Lyon, then it would be about seven-and-a-half hours, six-and-a-half with PACA.
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Finally, if trains could run through the night...
You could run sleepers from Lille.
Build some nice restaurants in Lille-Europe, run a shuttle service to Lille from London and from HS2 stations (when the link is in place) and then have sleepers leaving Lille at 10-11pm.
On current timings (ie the fastest daytime train on these routes), before there are any more line upgrades, a 22:00 departure could arrive as follows:
Vienna 08:08, Budapest 11:08
Copenhagen 09:09
Berlin 05:47, Warsaw 11:15
Prague 08:49
Madrid 10:21 (via Valldolid, and I know that's impossible because you'd need a gauge-changing TGV)
Barcelona 06:18
Bologna 06:39, Florence 07:23, Rome 08:45, Naples 09:55
Milan 05:34, Venice (Mestre) 08:09, Trieste 09:53
Obviously many of these are being speeded up.
Coming back, most of these have reasonable departure times for, say, an 07:00 arrival, which would give you time to clear security in Lille and board an 08:00 shuttle arriving St Pancras at 08:30.
Offer breakfast on the shuttle and dinner in Lille before departure so you don't need a dining car on the sleeper.