Consider each opportunity you might apply to on a case by case basis, fairly, and by playing to your strengths!
Some like Accenture only seem to look at you if you have a 2.1 - first degree (presumably from just about anywhere!).
Others take a more rounded view, so if you have some real experience in investment banking, set up Microsoft in your garage, or hold a Phd from Oxbridge, then they would still see the merit of your application.
Personally I'd apply to consultancies with a good mix of skills on the team. It tends to make for better teams (so if it's not your strength it's likely to be someone else's, and if it is a commercial question then you can offer something in return).
Real commercial experience does not always seem to get the coverage it should do these days. I can understand why it might not fit with a public sector consulting culture, but when there are private sector clients involved it seems a bit misguided to ignore someone with significant commercial experience.