Final answer:
The Beatles began filming their rehearsals at Twickenham Film Studios in January 1969, which were later showcased in the 'Let It Be' documentary.
Step-by-step explanation:
The facility where the Beatles started to film their rehearsals in January 1969 is known as Twickenham Film Studios. These sessions were part of the project that ultimately became the documentary film 'Let It Be.' The objective was to capture the creative process of the band as they prepared for what was to be their final album at that time. The rehearsals are notable for showcasing the band's dynamics and the eventual tensions that arose during the creation of 'Let It Be.'