It would unfortunately be enormously disruptive to convert lines to modern UIC standard loading gauge. Even a tehcnically isolated passenger operation such as the london Tilbury and Southend (LTS, operated by c2c, the DfT call it "Essex Thameside") would be massively difficult.
One way would be to replace platforms with "compliant" ones, then add a temporary structure on top (as was installed on the Statford International international platforms during the Olympics). Do that to the whole route, make sure stuctures are cleared as well.
Meanwhile build, test and fully run in a complete new fleet. Send drivers to test track for training. have them delivered to a depot connected to the line.
One weekend, shut the entire operation. Remove every temporary platform top. Remove the old stock. Introduce new stock. Reopen. hope nothing breaks down.
Now, that could maybe be done with the LTS. No other passenger trains operate onto it. Freight will be able merrily pass through the reopened line.
Any other line, it gets more difficult. Take say the Greater Anglia West Anglia Main Line and Lee valley lines.
Fine, you could probably repeat the trick somewhat. You could even designate specific platforms at Liverpool Street- the new stock would be banned from any other, and GE trains would not be able to safely use the new platforms in passenger service.
Even at Stansted, it wouldn't be so bad, XC use a different platform to Stansted Express. They'd have to stop stopping at Audley End though.
Cambridge though? A potential nightmare. plenty of space to build the complex station you'd need. You'd almost certainly have to end through-running to Ely and Kings Lynn, with all WA services terminating in their own platforms that GA rurals, FCC and XC would be unable to use. The WA trains would not be able to use other platforms, just their own. They could not even tun ECS through other platforms.
At the London end, the WA trians would be unable to access Illfrod until the GE was imilarly upgraded.
And that's a simple example.