In the UK Manual of Air Traffic Services there's a bit that says
When a pilot has given certain items of information normally associated with an emergency message but has not prefixed the transmisison with 'MAYDAY' or 'PAN', the controller is to ask the pilot if he wishes to declare an emergency. If the pilot
declines to do so, the controller may, if he thinks it appropriate, carry out the necessary actions as if the pilot had declared an emergency.
Although Doc 4444 and the like only cover emergencies in brief. once an emergency has been declared the pilot is getting the benefits of the Alerting Service - if you search on that topic you may find more info.
As for what the pilot gets - especially if it's a MAYDAY, the pilot can expect to get priority handling or whatever he/she wants, perhaps a discrete quiet frequency, to be met with emergency services if appropriate, any information that is requested as quickly as it's possible to get it and whatever else the controller can think of that might help. At least, that's how it should be - although the resources etc that the controller can call on do vary significantly from one place to another.