The first generation Eurostars (The so called TMST sets) were build to UK loading Gauge. They could run anywhere in the UK, and that is why, when the Regional Eurostar did not materialise some of the sets intended for that were used on the ECML.
You will notice if you have a closer look at them that they have "narrower at the bottom" profile that is typical for UK rolling stock.
Then newer Eurostar sets (the e320) are Siemens Velaro sets (similar to DB's ICE-3) and these are built to (Continental) European loading gauge, and thus cannot circulate beyond HS1, and the spur to Temple Mills.
Trenitalia intends to use FR1000 trains, and these are build to European loading gauge as well. Trenitalia has a bunch of those FR1000 already, and has more on order. That is why they are currently the only real contender to compete with Eurostar. Contrary to all the others that have announced plans they actually have trains.