If you can get back to a Post Office today and have access to a variety of pieces of information about your car you can get a tax disc using a form that’s specifically for when you can’t find your V5 or renewal notice. Sadly I can’t remember what code it has off the top of my head but there is such a thing.
If you have your V5 to hand you can use that to renew your tax, you don’t need the reference from the reminder.
I don’t know how much slack the DVLA will cut you if you don’t renew it in time. Last December I failed to renew mine in time but it quickly became apparent that the DVLA had messed up the processing needed when it lost its previous owner’s custom plate so their systems were probably not actually aware that it was untaxed until I started asking them why I couldn’t tax it…