I believe GSM-R login is also required at Paddington, such that the train's reporting number is properly represented in the 'berth' — otherwise the signaller cannot set a route out of the platform.
No, you have the logic backwards - the signaller being physically able to set route is in no way related to GSM-R setup at either end of the route. The driver has to input the correct/matching reporting number into GSM-R as is on the train describer, otherwise it will not register properly (it will come up “check headcode” or similar if wrong).
GSM-R login is required almost everywhere on the rail network, so that you can correctly receive and send calls to the signaller, including correct routing of emergency calls.
The signalling at T5 (ECTS) is quite different to that at Paddington (TPWS/AWS). For ECTS to work the train has to be able to receive a GSM-R data signal with the Movement Authority. No signal or poor signal means there is no Movement Authority. The issues with GSM-R at T4 were really bad and special measure had to be taken for it to work. GSM-R operating frequency is very close to the normal mobile phone frequencies and that can be a cause of interference I believe.
ETCS - European Train Control System
Loss of communication or poor signal will be indicated as such on the driver’s display, along with exceeded authority and full brake demand. This happens from time to time, but for the most part the issues nowadays are thought to be at a software level between the train and the RBC (Radio Block Centre).
HEx do anecdotally have most of their problems on setting up or starting off from T5; then software glitches while running under L2 (resulting in loss of Movement Authority etc); then fail to transition from L2 to NTC (National Train Control, AWS/TPWS). Fail to transition NTC to L2 seems very rare.
Whereas anecdotally for Crossrail I’d say most of their issues are the random glitches I referred to above for HEx, followed by fail to transition NTC to L2; followed by setup issues at T5; with fail to transition L2 to NTC is seemingly rare for them.
That both operators are having very similar sets of issues with very differing on train software architecture points to infrastructure not being quite up to scratch, if it were the trains I’d expect one or other architecture to work significantly better than the other. But they’re both failing about the same amount, in broadly the same ways - HEx have “caught up” Crossrail in only having a couple issues per day, compared to a dozen or more per day in the first weeks of operation.
There’s not been the extensive operating to T4 by either operator yet to judge if the known GSM-R signal issues have been resolved. That’ll come soon when HEx go back to 4 trains per hour.