Hi all. Well it took Pimlico 7 visits and Vaillant 3, but the boiler is working again. Vaillant replaced the pump Pimlico installed, as it wasn't Vaillant branded. When that didn't work, they replaced the thermistors and any other connectors in the system (sorry, can't remember the names.) When that didn't work, they replaced the PCB and the adjusted the gas intake valve. The problem disappeared after the last visit, so hard to tell if it was the PCB or gas intake valve. Pimlico had adjusted the valve early in the process based on Vaillant's recommendation for a different problem, so it may have been that.
In the end, I have had virtually all, if not all, replaceable parts of my boiler replaced. I'm happy to say I have now had hot water and heating for the past 3 weeks with no trouble.
Thanks for all the suggestions.