Well, "MI" is an artifact acronym dating back to the Directorate of Military Intelligence, which was part of the British War Office. The original organization that eventually became the DMI was the Department of Topography & Statistics, founded in 1854. Different sub-organizations weren't called "MI#" until WWI. The DMI was absorbed into the Defence Intelligence Staff (DIS) in 1964, when the War Office was subsumed by the Ministry of Defence (though some parts of the DMI's duties were transferred to other parts of the MoD outside the DIS). Today, only MI5 and MI6 remain, and the "MI" titles are not actually the organizations' titles anymore, but rather just colloquial terms for the Security Service and the Secret Intelligence Service, respectively. A listing of the various "MI" departments that existed at various points can be found on this page; the only two numbers that never existed outside of fiction are MI13 and MI18. If the British had a UIU counterpart as part of the DMI when it existed, my suspicion is it would have been part of MI10, which handled weapons and technical analysis during WWII. That organization was one of several that got absorbed after the war into the Government Communications Headquarters, or GCHQ, which today is the British counterpart to the American NSA, responsible for SIGINT, COMINT, and ELINT. Whether a hypothetical British department for paranormal analysis (or whatever one might call it) would have made the transition with the rest of MI10 to the GCHQ or spun off either into its own organization or been absorbed by some other part of the British government could really go either way, depending on how an author chose to write it. Both such eventualities would be plausible to me as a scholar of intelligence history.
That said, it would have been equally plausible if the British counter-anomaly organization were its own organization, independent of the military or intelligence apparatuses since its inception (which could conceivably have been hundreds of years ago). If that were the case, there would undoubtedly have been bureaucratic turf battles between that organization and the military and intelligence communities (in as much as those communities were aware of the organization's existence), since government organizations as a general rule never get as much money or bureaucratic control as they would like.