Lothian did withdraw a few services in the West Lothian area not long after McGill’s took over, but that may just have been coincidence?
Lothian had to register their intention to withdraw services with the relevant authorities, which I believe is 42 days, I may be wrong. This doesn't include the internal discussions and planning within Lothian to come to this decision, so this adds on another month or two at least. These changes had been planned and brought about, due to the fact that they were suffering from a shortage of drivers, a consequence of Covid, and were constantly having to cut short journeys or cancel them altogether. The service they were providing at that time was woeful, so they took the decision to consolidate and concentrate on their Edinburgh corridor, so withdrew the two West Lothian services where McGill's had a presence, thus ensuring that there was a service being provided to the effected communities.
The news, when it came out, that McGill's had bought over First's operations in the East, was a surprise to most people. I therefore agree, that it was purely coincidental regards the timing of the withdrawals.