After putting the (top of the) list of largest metropolitan areas on a map, and finding the distances and estimated travel times, this bit takes on a whole new meaning:

One problem for the UK is population density. The technology is straightforward enough - providing it's not Maglev - but the local politics are extremely complicated. You can be sure that whatever route is chosen there will be protests, court actions, and other legal issues which will create huge delays.

I suspect other parts of Europe may have similar issues.

The 10 largest metropolitan areas in Europe form a connected network except for Moscow and Istanbul. However, the list of metropolitan areas largest than 2M people includes Kiev, Bucharest and Krakow, and that is enough to connect the whole network.

Now, the issue of right of way is important. Basically, the 3h leg limit corresponds to roughly 900km, and it's hard to find 900 km in Europe without a largish city in the middle that will demand that the HST stop in them. Many European Countries are not even 900km across, so one could even find a national capital in between two nearest neighbours of the network.

Many Spanish regional governments insisted that the AVE had to go through some of their provincial capitals in order to collaborate with the Madrid-Valencia, Madrid-Sevilla or Madrid-Barcelona lines. In building a pan-European network, the national and regional governments would probably bargain similarly.

"It's the statue, man, The Statue."