When you book your driving test, your booking is made with DVSA (Driving & Vehicle Standards Agency) rather than DVLA (Driver & Vehicle Licencing Agency). It’s an odd fact that while the DVLA are responsible for driving licences, it is DVSA who are responsible for the driving test to enable you to get a full UK driving licence.
If you want to change your driving test date, your first step should be to go online at the DVSA part of the gov.uk website and see what dates are available. If there are no suitable dates free at the driving test centre of your choice, then you should contact a driving test cancellation finder service to get you an earlier test. When you pass your test the examiner will offer to send off to the DVLA for the new full driving licence on your behalf. If you choose not to take up this option you will need to apply to the DVLA yourself for your full licence which you will have to do within 2 years of passing your driving test. If you do not do it within 2 years you will need to re-take your driving test.