Agree with Nulli Tertius on this one.
I have met loads with strong regional accents at large commercial firms. Doesn't seem a problem. Where it WOULD be a problem is with somewhere like Withers, and especially Farrer & Co. They're big private client firms and their clients can be exceedingly 'posh'. I know a partner who was originally at Farrers; a client of Farrers; and also someone who's friends with a partner there and they all give the same impression. My friend even said when he was at a meal with a partner there the partner monitored how he used his knife and fork!