This one's reasonably straightforward to answer (based on my years as a strategy consultant before leaving to set up Top-Consultant).
As strategy consultants tend to be providing Board level advice and working with senior management, their "client site" tends to be the client's Headquarters office. If you think of major blue-chip firms then this is typically going to be in the capital cities of major countries. For most industries this means you are likely to be working on London projects some of the time (for London headquartered clients) and then overseas (Paris, Munich, Lisbon, etc.) for the rest of the time.
Travel with strat firms is quite often "glamourous" to the extent that you tend to go to nice places - though be warned that the inside of an office / hotel looks pretty similar wherever you are in the world!! In my strat time I worked in London, Hong Kong, Johannesburg, Bangkok, Luxembourg city, Frankfurt, Zagreb.
By contrast, consulting work that is not strategic is more likely to be being carried out at the various office locations of the client. So non-strat consultants are more likely to find themselves trekking off to Newcastle, Manchester, Bristol, Cardiff, Birmingham, etc.
The only way to have a more fixed base in London is to choose a sector which has lots of important clients in London. If you work in the Financial Services / Investment Banking field for a strat firm, then more of your clients will be based in London - though you'll probably still find yourself jetting off to New York or Frankfurt from time to time.
Hope that helps. Tony Restell (Top-Consultant.com)