The ambulance service in the UK is run by the National Health Service (NHS).
There are a number levels of training currently available. If you wish to work on the 'front line', that is attending 911 emergency calls (999 in the UK), then it is necessary to complete the EMT course. This is currently approx 9-12 weeks and covers the clinical and driving aspects of the job, followed by 12 months on the road as a 'trainee' before gaining your qualified status.
If you want to be a full paramedic then a 3 or 4 year university course is now the main route to qualification, and competition for places is extremely fierce. A UK paramedic is an independent health care professional, registered with the health professions council (their website has loads of information). UK paramedics use their own clinical judgement to treat patients, within set guidelines. Direct physician control does not happen.
The fire and rescue service is completely separate and has no input into the provision of ambulance or EMS care. Have a look at www.fire.gov.uk