It sounds like you've answered your own question here. In industry, naturally very few people will have heard of the firm. Then again, a lot of people in industry aren't that familiar with the differences among consulting firms generally. If they are even considering hiring someone with a consulting rather than industry background, they will either be dead-set on someone from MBBB, someone from the Big 4, or will be hiring on the basis of personal reference (which is presumably where you will have the best chance).
For consulting firms in London, again I think your best chances of an "in" will be through personal reference. Although recruiters could verify PJP's credentials with a phone call, in reality the number of applications MBB recruiters see means it's not usually something they'd do at the first round of deselection.
My advice would be to discuss with PJP their networks and opportunities for projects in London. This will give you an initial idea of how easy it will be to make the transition.