Firstly, well done.
Secondly, you have passed your test so you can drive. However, you need to change your details anyway. You have 2 years from passing to change it which is daft, however do it asap.
If you're stopped by the police for any reason you'll get asked from your docs and if you can't produce your DL they'll likely issue at HORT/1 or 'producer' form which gives you 14 days to report to a police station of your choice to show your docs otherwise it's an offence under the road traffic act.
however, most traffic bobbies will understand this situation if you explained it road side and you'd probably get discretion if you weren't arsey about it. I'd certainly let it slide (i dont work traffic tho) as long as you could show me other ID to prove who you were - passport is best, that way they could run checks on you and they'll know you do have a full license. I'd photocopy both your pass doc and the prov.DL and carry those too before you send them away incase the DVLA take longer to update.
I think the DVLA say about 3 weeks for it but depending on demand it could be quicker or longer.
if you have any doubts then phone the DVLA (your best bet) or drop into a police station and ask to speak to a traffic bobby who will tell you 100%.
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