Theoretically possible in the sense that the lines exist and are connected? I don't know for sure, and while it may be a very convoluted route, I wouldn't be surprised.

EDIT: This map[1] suggests you could do Norwich -> Birmingham New Street -> Bristol Temple Meads. Without having to go through London, I suspect this is indeed possible without changing trains, even if no operator runs that route.

[1]: https://www.thetrainline.com/cmsmedia/cms/9384/national-rail...

Perhaps, but if it means CrossCountry running the service you’d rather walk

Theoretically: Absolutely - If Norwich has the same type of electrification as the Great Western Main Line. The Class 345 trains that run in the Elizabeth Line, have the signalling for the Great Western Main Line, Crossrail Core & Great Eastern Main Line.

In practice, no - different train companies operate all 3 lines

The train companies operate trains, not lines. For example; CrossCountry’s service runs on track shared with other passenger companies and freight from Penzance up to Scotland.

The train companies (in general) operate a franchise for a particular region of the railway (including mandatory service to less profitable areas in the network). CrossCountry is a quirk in that it operates the "Cross Country" franchise and it has no specific track to itself. Eurostar, Grand Central and Heathrow Express are Open Access operators in that they have no franchise of their own but pay to use the track for each journey. The whole thing is needlessly complicated to be honest, there's no real exclusivity - but there's massive capital costs involved in setting up a competitor and more risk involved for these Open Access operators.

Great Western Railway operates the Great Western franchise which incorporates much of south wales and south west of england - Including the Great western mainline. Abellio Greater Anglia operates most services on the Great Eastern Mainline

As for permissions I don't know if the crossrail core section has exclusivity granted to TFL (or its subcontractor MTR Corporation), given it's run as a concession this may be the case.

Currently the Elizabeth Line trains are slow and stop everywhere. Think of tube rather than train.

From the West end of the line in Reading nobody takes the Elizabeth Line to go into London, for instance, it's much too slow compared to the direct fast GWR trains.

This also means that the line probably cannot be shared with fast trains for direct routes from East to West, I think.

I live near Ealing and it's been a tremendous improvement to my commute.

You're correct, fast intercity services and the stopping crossrail services likely cannot coexist.

But technically it is possible, assuming the requisite signalling standards are met by the rolling stock intended to be used. It's possible that only the Class 345 and Thameslink's Class 700's can meet this requirement currently.

You don't even need Crossrail for that: https://en.wikipedia.org/wiki/London_Crosslink

great eastern from norwich, go round half the loop at stratford, join the north london line and exit onto the great western at acton

freight trains will be doing that complicated bit every day

Back in the early 2000s there was an occasional service (Anglia Crosslink) that ran that way. Ipswich to Basingstoke was the longest route.

It would, but it would be unwise. A commuter/metro service has quite different needs from a long-distance service (this is one of the lessons of Thameslink, which does work that way, with trains running from Brighton to Peterborough through central London, and attendant compromises). The Crossrail plan is to divert the commuter services through the tunnel and free up more space in the terminals (Liverpool Street and Paddington) for the fast long-distance trains.