I didn't realise the joint line was closed on Sundays, I always thought after re-signalling that it was controlled by a box that was open 24/7 (Lincoln?) but looking at real times, nothing is scheduled on Sundays (not even freight) so I can see how this could be a problem. That will also explain why in EMRs mega timetable masterplan Lincoln - Peterborough isn't planned to get a Sunday service (whether the EMR masterplan ever happens is another debate)
In the past limited depots signed the joint line, so on the engineering weekends I assumed that drivers changed at Peterborough and / or Doncaster to ensure that the maximum number of trains could use the joint line. I do remember years back on the 0930 Edinburgh - London (which was a HST back then) we had wires down at Claypole, but needed to terminate at Doncaster as although the HST was fine, the driver didn't sign the route. I assume that if say 10% sign, they diagram them on the weekends to ensure they are pretty much shuttling the joint line, whereas I assume on a normal Sunday you will have a lot of train crew that don't sign it that could be caught up.
On other ECML closures HT have diverted, LNER and GC less so.