This topic has been discussed before and you might find some useful information if you search for earlier threads.
Your question relates to an incident in the UK and most of the answers are specific to the UK; other countries do some of this stuff very differently. I am long out of ops but back in my day the controller had quite a lot of discretion in the response to an aircraft in distress or emergency (you can read that as there wasn't much written down about what problem got what response, just a few examples for guidance). We have now moved into an environment where a lot of controller actions are pre-ordained - from day one of training, controllers are often taught if 'this' happens, you do 'that' - there may be a reflection of that in zonoma's reply.
In the UK my understanding is that still, with respect to downgrading, once an emergency response has been initiated at an airport, which technically can only be done by ATC or RFFS, downgrading the level of response can only be done by the Fire Officer i/c, and in most cases that will only happen after the aircraft has landed (again, things happen differently in other places). The problem with downgrading is that if events unfold in a way that mean that had downgrading not occurred the outcome would have been better, whoever took the decision to reduce the response will have some very difficult questions to answer.....so even if the pilot declares that there is now a situation which warrants a PAN, it is unlikely that the ground response will be downgraded before the aircraft is on the ground.