Having just researched this, here's some tips. Filming in London as a professional
isn't quite as easy as it seems. London is covered by a patchwork of authorities. It's all explained here:
There are three main authorities:
Westminster City Council
Shooting Buckingham Palace from the Mall side is covered by Royal parks.
Parliament Square is GLA, however there are some areas of pavement in the
Square that come under WCC authority. For Trafalgar square Prince Charles'
traffic Island (the one in Charing X direction) and the North Terrace (pavement in front of National Gallery) are WCC, the square itself is GLA. Piccadilly Circus is WCC.
Whew. Complicated. Well, not that complicated if you plan where you want to
shoot and then check the locations at the filmlondon website.
Getting permission takes 3-5 days depending on the authority, and
generally requires a fee being paid. Royal Parks do a yearly news permit for news
crews which is cheaper, WCC can be free depending on the production. You may be
required to clear it with the Police afterwards.
All the phone numbers are on the filmlondon website. Good luck!
The first paragraph of which seems to clarify everything and
cover any of us just doing filming that doesn't involve laying down
non-portable equipment. I'd suggest printing it off and keeping it at
hand. The site also has a map of the area City of London covers. So worth
taking a look at that also.