While CBTC signaling is being installed, the original fixed-block signaling remains in place. It's nearly impossible to do this otherwise - you'd have to shut the entire line down for the entire time it takes to install new signals - which could be years.
Once the new signal installation is complete the two systems may operate in parallel (with the fixed-block system acting as a possible backup if CBTC fails, and also so older trains can still run).
The issue is that with 24/7 operation, there is no consistent regular time where work like this can be done. It can only be done in short spurts (late-night shutdowns) or the work crews in-lined with train service, which has obvious safety concerns as well as drastically reducing the capacity of the line.
As you point the integration work for this is complex and is usually the primary driver of delays and cost increases.
I can completely sympathize with the difficulty of upgrading 24/7 critical infrastructure, especially given that mistakes can kill or injure people. I'd try to just deploy separate parallel systems instead.
At a comically reduced scale, businesses migrating between ERPs (or similar line-of-business apps) often run both the old and new systems 'side-by-side' for some period of time. What I've never seen or heard them doing is trying to replace components or sub-systems of the old system with those of the new system. That kind of integration work would be awful and would probably drastically increase the odds that the migration would crash and burn.