The simple answer is yes! The only issue you may have is £2000 may well be over their 'without notice' limit. I.e you may need to give them 24hrs notice so that they have the cash ready for you. You could be faced with this issue even if you turned up with your card. With Santander, you do not need your bank card to deposit, withdraw or even close an account, providing you have photo I.D
I am 100% sure that you will be able to withdraw the funds, however if for any reason you cannot, you will be able to make a chaps payment to another bank account in the bank using just your driving licence. Maybe ask a friend or relative if they wouldn't mind you transferring them the cash and then get them to withdraw it for you. Though again, you won't even need to resort to that.