Would Eurostar come under nationalisation, and if not, who pays to maintain their tracks etc? What about Merseyrail, is that classified as being under British Rail or it's own local Metro service (like Metrolink)?
Eurostar is not owned even partially by the government - effectively it is open access controlled by SNCF.
HS1 is government-owned but its capacity is sold to a private consortium of mostly foreign pension funds until I think 2040, when it reverts to HMG.
NR maintains HS1 under contract to the owners.
Heathrow Express (the branch off the GWML and terminal stations) is owned by Heathrow Airport, with infrastructure managed by NR under contract to them.
Rolling stock provision is contracted out to GWR (DfT)/Elizabeth Line (TfL).
Service at other airports (eg Manchester, Stansted) is, I think, provided by the regular DfT TOC/NR system.
Merseyrail is let as a concession to Serco/Transport UK until 2028, and is not under the purview of DfT though it is a railway TOC (as is eg TfW).
The infrastructure is part of NR, the rolling stock is owned by Merseytravel and maintained by Stadler.
Unpick that lot if you will!
Metrolink, like other tram systems in the UK, is not a DfT TOC, being run by TfGM, and its infrastructure is not part of NR.