When I did that
timeline of British Isles titles over ten years ago now(!), I included an appendix of notes on federation changes, which might answer most of the questions related to that, but would need updating with any additions or corrections (there were some corrections in the threads back then). The one possible further addition of federation changes needed is William Albert Fairhurst (from Ian Thompson's post above). Fairhurst is in the original list (well, after Leonard Barden and others pointed out various missing people), but I was not aware that Fairhurst changed his federation - are there other federation changes from the early years (1950s, 1960s and 1970s) that might have not been noted here?
*(i) Dimitri Anagnostopoulos (previously known as Demetrios Agnos, name change took place in 1996 when he switched from ENG to GRE around the same time as he gained the GM title, so it is not clear which federation the application was made by, but when he first appears in the lists, it is as a Greek GM, not an English GM; He did gain the IM title while registered for England)
*(ii) Ketevan Arakhamia-Grant (nee Arakhamia, now Arakhamia-Grant; gained WIM and WGM titles while registered for USSR, code changed to Georgia in 1992, the switch to Scotland under the name Arakhamia-Grant took place in 2008, and her GM title was while registered for Scotland)
*(iii) Alexander Baburin (registered as Aleksandar Baburin for URS when he gained the IM title, the code changed to RUS in 1992, with the name change and federation change to IRL taking place in 1995; the GM title was awarded a year later while registered for IRL)
*(iv) Jana Bellin (nee Malypetrova, then Hartston (1970), later Miles (1978), then Bellin (1988); born in Prague, CSR, was registered as ENG from at least 1972 onwards and won the 1970 British Women's Championship; she gained the WIM title in 1969 while registered CSR; she gained her WGM title while registered for ENG)
*(v) Murray Chandler (gained his IM title while registered for NZL, switched to ENG in 1982, gained his GM title while registered for ENG, switched back to NZL in 2007)
*(vi) Dagne Ciuksyte (gained the WIM, WGM and IM titles while registered for Lithuania, switched to England in 2007)
*(vii) Joe Gallagher (gained his titles when registered for England, switched to SUI in 1994)
*(viii) Rita Hennigan (nee Zimmersmann, later Hennigan (1997), and later Atkins (by around 2010); was registered to HUN when she gained the WIM title and switched to ENG in 1997)
*(ix) Tony Kosten (registered for England when he gained his titles, switched to FRA in 2002)
*(x) Bogdan Lalic (gained his titles while registered for Yugoslavia, code changed from YUG to CRO in 1992, then CRO to ENG in 1999, and changed from ENG back to CRO in 2005)
*(xi) Susan Lalic (nee Walker; later Arkell, 1986; then Lalic, 1994)
*(xii) Jacques Mieses (born in Germany, moved to England in 1930s, was on the list of inaugural grandmasters in 1950)
*(xiii) Nick Pert (GM title year possibly 2004, FIDE says 2001, personal website biography says 2003)
*(xiv) Gulnara Sachs (previously Gulnar Sakhatova) (gained her WIM title while registered for URS, switched to KAZ in 1992, and then to ENG in 1993 at same time as name change)
*(xv) Anne Sunnucks (nee Patricia Anne Sunnucks, now listed as Anne Mothersill, "earned the Woman International Master title by placing second in the 1954 Western European Zonal", name change was around 1988)
*(xvi) Simon Webb (gained IM title when registered for ENG, later moved to Sweden but continued to be registered as ENG)
*(xvii) Gary Lane (gained his IM title while registered for ENG, and switched to CAN and then AUS in 2001)
*(xviii) David H. Cummings (gained his IM title while registered for ENG, and switched to WLS in 1995 and then CAN in 2000)
*(xix) George S. Botterill (initially registered for ENG, but gained his IM title while registered for WLS; switch from ENG to WLS took place in 1974)
*(xx) Ingrid Lauterbach (gained her WIM title while registered for GER, switching to ENG in 1993)
*(xxi) Jacob Aagaard (obtained his IM title while registered for DEN, switched to SCO in 2007, obtained his GM title while registered for SCO, then switched back to DEN in 2009)