Public Filming

"If you are a small crew — five people or fewer — using a handheld camera and your filming will not cause an obstruction then there is no restriction to filming on London’s public highway. In some boroughs this also extends to small crews with a tripod. No licence or any form of official permission is required.
Common law rights allow users of public highways to “pass and re-pass ... and to make reasonable use of it”. Section 137 of the Highways Act 1980 restricts these rights where the use of the highway causes an “obstruction” to other users rights of free passage. If you are considered to be causing an obstruction then there is a risk that you will be stopped from filming.
If you think you are likely to cause an obstruction then you need to apply to film via the appropriate Local Authority, however we strongly recommend that you always inform the relevant Borough Film Service if you are filming on their streets. If the local authority is not aware that you will be filming in the area then you can be asked to stop. If a member of the public complains and you don't have permission to film, then you may be asked to move on by police. Some Local Authorities provide Notices of No Objection for small crew filming."
Source: http://filmlondon.org.uk/get-permission-film

From this research, we have discovered that we will be able to film in public without an issue.