I don't think much of the attitudes of the insurer or the broker.
Lets start with the broker. A brokers job is to come up with a competitive quote for his client's insurance needs come renewal time. The company is irrelevant as brokers usually, unless they are tied brokers, represent a number of insurers and get can get quotes from them all.
The insurers owe their client, your mother, a renewal notice including her NCB, a significant time, I think it's 15 days, before her renewal lapses, to allow her to get quotes from other insurers.
I'd contact the insurer saying the broker no longer sells their car insurance product and demand the NCB certificate immediately.
I'd also contact the broker and telll them to instruct the insurer to issue the NCB to them for forwarding to you.
If it's possible, ask your mother to attend each place in person and demand to speak to whoever is in charge.