As mentioned HS1 is signalled using TVM430, a now outdated French in-cab signalling system with fixed block sections. Conventional Signalling was necessary in this last kilometre from the tunnel mouth to St Pancras station due to the National Rail interfaces, the need for routing information and because short block sections are required to meet the specified operational headway/journey time (which cannot be achieved within the low speed codes available to the TVM system due to the speed profile of the layout). However with ECTS this wouldn't be the case.
The Thameslink Core Route, London Bridge - Kentish Town/Belle Isle Junction, has ECTS Level 2 (in-cab signalling with fixed block sections). This has no issue with coming off conventional signalling to ECTS anywhere along the route. Although there is Conventional Signalling along the Thameslink Core Route, a train will only be using one of the systems. When in ECTS, the Driver ignores Conventional signals. There are also more block sections with ETCS than with Conventional Signals, so it is possible for a train in ETCS to move into an already occupied Conventional Signal Block Section, because of ETCS Block separation.
So yes Curzon Street will be signalled with ETCS. I suspect Level 2, but ideally there is an opportunity to try the first Level 3 which has moving block sections, but knowing the UK, we will be behind in modernisation. In fact I wouldn't be surprised if they installed Semaphores as they seem very reluctant to get rid of this nineteenth century technology.