There is actually no precise equivalent of the FBI in the UK: MI5 has a similar National Security remit but is an intelligence (strictly speaking, counter-intelligence) agency while the FBI is largely a police agency that also deals with intelligence-related National Security issues.
The UK also has Special Branch which, like the FBI, has some national security remit and is a police agency however, unlike the FBI, it's not national. Each (regional) police force maintains it's own Special Branch, though the various Branches do co-operate .. in theory, at least.
I believe there are those in the US who believe the FBI should be split into separate entirely police and entirely intelligence agencies. There are probably people in the UK who think MI5 should do it's own policing.
The UK Freedom of Information Act does include a right to request a copy of information held about you by the police (for a £10 fee) however, this is subject to a number of constraints and exclusions, for example any information that "would be likely to prejudice the prevention or detection of crime". More information is available here
Apparently, the FoIA also applies to MI5, but their webite
only really talks about the exceptions - basically, anything National Security-related is excluded .. and their remit is National Security. So good luck getting anything there