Yes, you will still be able to register with a GP. There are accurate reports that focus on the state of the NHS being in trouble due to lack of funding. This mostly effects wait times within the internal systems.
Dependant on your borough in London or UK wide, you will still be able to find a GP practice in your area that will register you as a patient and give you an NHS number. You may find that you will be assigned a GP practice that is slightly further away from your address.