Typically, how many days do Bain consultants travel per week (if at all) at their London Office?

Unfortunately the answer is really a "it depends". To give you a rough idea:

• Roughly 1/3rd of cases are in London (Bain or Client office) - no travel
• Roughly 1/3rd are within M25 / <1 hr from London on train / taxi
  • A lot of people still choose to come back to London every day and commute
  • Travel days vary completely - Monday-Thursday, Tuesday-Thursday, Monday-Wednesday are all possible
• Roughly 1/3rd are travel cases (either long distance train or flight)
  • Travel model also vary: I'd say half are Monday-Thursday, and half are only 3 days a week (Mon-Wed or Tues-Thurs).

In terms of choosing not to travel or not - if e.g. you have a good reason to want to stay in London (e.g. young kids) they generally try to accommodate. However, this can't be guaranteed and you may still be required to travel. Overall, I'd say less than a quarter of the office is actually completely out of London for longer than 2 nights a week.

As a separate comment, this also varies a lot by practice. For example, Private Equity consulting has almost no travel (95%+ of cases are in London).