Frustratingly Google maps gives London to Paris and London to Newcastle *by road* both as 288 miles!! (give me Geordieland anyday!!)
Which is further by rail isn't clear, but it's likely to be close!!
The European Rail Timetable gives St Pancras-Paris Nord on Eurostar as 492km, with King's Cross-Newcastle at 432km.
The Eurostar route via LGV Nord goes the long way round via Lille, a more direct route (following the autoroutes) goes via Lens and is a good bit shorter.
Paris-Bordeaux is 570km, Paris-Marseille is 750km, both of which are run non-stop several times a day.
I once used a Paris-Nice TGV which was first stop Toulon (817km, 508 miles, avoiding Marseille).
That's about the same as London-Edinburgh-Stonehaven.
Today's similar trains call at Avignon, which breaks the long non-stop run.
In Spain, Madrid-Malaga on AVE is 513km, Madrid-Barcelona is 621km.
In Italy, Milan-Rome is 568km.
All run non-stop on high-speed lines.
The Germans tend to stop at major cities en route, so don't have the same long non-stop options.
PS Having said that, there are a couple of fast services between Frankfurt and Berlin which is 559km to the first stop at Berlin Spandau.
Anyway, we seem to come 5th or lower in the European non-stop league, but possibly top where the route is a classic one.
I'm not sure that is a record to write home about...
Sorry, just discovered Gothenburg-Stockholm is 455km and is run non-stop a couple of times a day; this is a classic line, and longer than Newcastle-King's Cross.