BuddyPress is compatible with a WordPress Multisite setup. There are several different ways to set it up.
By default, BuddyPress data is stored in a networkwide way. Generally, that means that, within the network of sites, only one site is dedicated to BuddyPress features/content – you might think of this as the “community” site within a larger network of blogs. However, it doesn’t sound like this is exactly what you’re looking for. So there are a few other options.
The first option is a BP flag called BP_ENABLE_MULTIBLOG. When enabled, BuddyPress content can be viewed on any site on the installation – such as ldn.example.com, nyc.example.com, etc. However, the content is still stored in an installation-wide way. That is, while you’ll be able to view groups, profiles, etc on many different sites, they will all be the same groups, profiles, etc. You can read more about BP_ENABLE_MULTIBLOG on the BuddyPress Codex, where I recently tried to spell out the use case (it’s an often-misunderstood setting): http://codex.buddypress.org/bp_enable_multiblog/
If you want to have separate BP content from site to site – so that London groups/profiles/etc are listed separately from NYC groups/profiles/etc, and so on – there is a second option called BP Multi-Network. This is a fairly new feature in BuddyPress (since BP 1.5), and requires some additional network plugins (one to create multiple networks, like Networks+ http://wpebooks.com/networks/, and one to enable the BP Multi Network functionality within these networks https://wordpress.org/extend/plugins/bp-multi-network/). Read more about this functionality here: http://wpmututorials.com/news/new-features/multiple-buddypress-social-networks/
If you want something a bit more sophisticated, where some BP content (like, say, Profile data and Friendships) are shared between sites, but other stuff is site-dependent (like Groups), it can be done, but will take a fair amount of customization. The setup I just described could be achieved through a combination of BP_ENABLE_MULTIBLOG and some judicious use of group metadata (so that certain groups are tagged ‘London’, etc). Here is a plugin that may get you somewhere with this idea: https://wordpress.org/extend/plugins/buddypress-group-tags/