I have met and worked with several PhD's at McKinsey and BCG. All of them joined at associate/consultant level (i.e. not entry-level trainees) fresh out of university. If you wish to join as a trainee do so now. Do not expect to join as a trainee with a PhD, they will expect the maturity to work at more senior level.
A few things worth mentioning. This applies only to the Strategy Consultancies (far less so for operational consultancies as was said above). VC/private equity firms will not value a PhD except if it is in Finance. Even then you won't get much value out of it. Strategy consultancies appreciate PhD's, assuming they are from top universities and in relevant subjects. The colleges you mentioned fit the bill. As for subjects, social science is a bit iffy. Business or a science would serve you better, i.e. math, chemistry, biology etc.
Also: Your pitch will make a big difference. Show up at the door saying, '...umm, I have a PhD and I'd like a job please...'will get you the door slammed. Show up and say, I have a PhD in chemistry and have a lot of insight to offer to your clients in industrial chemicals, materials and manufacturing, and you will attract interest. Similarly for whatever subject - show that you've thought about how it fits with the firms clients and how they will be able to sell you to them (that's what it's about bluntly).
As far as industry is concerned, a PhD is a handicap. No two ways about it. The only exceptions are CxO's who got there either through the technical route (i.e. Engineering, or possibly via careers outside the UK, where education is seriously valued).
Finally remember there are many ways into strategy consulting, as long as you can demonstrate the thinking and mindset necessary – it really is the most flexible of professions, but the other fields you mentioned (VC/PE) are far more narrow. They do sometimes take people from strategy consultancies though.