lol some interesting and mixed advice,
tomthumb Dont ignore this, im assuming the dealership did a licence check or you had a named driver with you with a valid licence as they should never of let you drive off without it, its not a case of a few more days/week wont matter or the “plod take no notice” it can invalidate your insurance and make you liable (fault or no fault) it is very unlikely but possible, it can also carry a fine upto £5000. if its not possible to renew online ( i think they have a within 30 day or 3 month rule or something) then my advice is to send it via the post office where you can pay a few quid for them to pre check the application, also do a “licence checker” with the DVLA or what ever its called this week, this will give a number valid for i think 21 days showing your upto date licence details and handy to have with you whilst awaiting a renewed licence. just in case.
it is nothing to worry about and a simple call to dvla will put your mind at rest . in the meantime do the licence check/share thingy. https://www.gov.uk/view-driving-licence, good luck & nice car