Becoming a amberlance driverWatch
This link should help. It is unlikely you will just drive an ambulance, though.
To work alongside a paramedic, you'd need clinical qualifications yourself because you will be doing patient assessment and treatment. There are various roles to choose, depending on how involved you want to get. You can become an Ambulance Care Assistant which has the lowest level of training and qualification. Other names for this can include Emergency Care Support Worker or Emergency Care Assistant. In some places, you have to be an ACA to be a PTS driver, and London has Non-Emergency Transport which uses ACAs so this doesn't necessarily mean you'll be driving emergency ambulances on blue lights. You'd need to check this when applying, but it should be stated in the advert.
You could also become an Ambulance Technician, also called Associate Ambulance Practitioner (except in London, where their AAP role is more like the ACA role), or Emergency Ambulance Crew. This requires more training because you'll be given more responsibility. Once qualified, you won't necessarily always work alongside a paramedic but would be responding to emergency calls and be driving with blue lights.
Whichever role you choose, the usual route is to find a trainee role with your local ambulance service. These will be advertised on the NHS Jobs website. You'd then get paid wages by the ambulance service and they will arrange your training so would pay the fees. You can go to university if you wish - I know NTU has an ambulance technician course - or can do qualifications through private providers. If you do it privately you don't get paid wages and have to pay the fee yourself, and then have to find a job afterwards.