As you may know the Eurostar was developed in fits and stages.
The French were bigger proponents of a trans chunnel link than the Brits and the development of the tracks for the Eurostar reflected this.
When the Eurostar started in 1994 the French side tracks were 300 km/h (185mph) at the start. The British side was max 160 km/h (100mph). The trains on the British side followed the classic routes from Waterloo to Dover. 3rd rail electric and congested with commuter trains.
The Brits had proposed a 300 km/h link following roughly the same path. Why? Well it had always been that route. Problem was 1) the route is very hilly. 2) the route goes through densely populated and affluent neighborhoods.
Nimbys everywhere and I don’t blame them. Construction at best would take a better part of a decade. Who wants to live with that? And the terrain meant lots of tunnels and/or deep cuts through the hills. Expensive. And the land to be acquired was not only expensive in itself but was full of homes worth about a million $ each. The whole project was just like the California HSR.
This is what a trip on the Eurostar would look like speedwise in 1994.
It looked like given the cost a proposed HSR line also into Waterloo on a similar route would be nevertime. Yet BR and government planners still stubbornly wanted the old route through affluent south London.
Finally Ove Arup who ran an enormous engineering and construction company in the UK proposed the British HSR to come in to London from the east (not the south) just along the north side of the Thames ending in St. Pancras (a terminal that traditionally went north to the British Midlands). This was unconventional! Novel! Brilliant! It was also cheap! Speed limit was kept to 230 km/h (140 mph) to keep costs down for the tighter turns in the congested area.
This area was delapidated. Think old warehouses and docks designed for sailing ships and long abandoned. The people living there were poor and often jobless. Rather than being nimbys the east Londoners were yimbys. Yes they loved the decade long construction jobs. And literally only a few homes were needed as the line went through unused industrial areas. And the land was basically free in many places. Win/win.
The new route was done in two stages. The first stage was easier to build and went from Dover on the Channel northwest through mostly open farmland (no thousands of million $ homes needed to be torn down there) to Southfleet just south of the Thames (about 25 miles east of London) where it linked in to existing tracks to Waterloo.
On the left side is what a Eurostar run from London looked like with a departure from Waterloo on existing route and connecting at Southfleet with the HSR.
The right side is a Eutostar run after the second stage from St Pancras on the north side of the Thames was completed to Southfleet.
This my friend is how to build HSR in the USofA.