I’ve still to watch the programme, 3Ts, but the IC125 is not and was not a tilting train. About the same time as the 125 was being developed British Rail was developing the Advanced Passenger rain (“APT”). This was a tilting train but it was not a success. Despite being instrumental in the development of advanced braking systems and electrical traction, overall it was never fully developed to its potential. A press trip from London to Glasgow broke the record for that run but the gentlemen of the press reported that its tilting mechanism made them very queasy. Some reports, however, suggested that this was a result of the copious quantities of free BR booze that Fleet Street’s finest had consumed before the train passed Watford Junction:
https://en.wikipedia.org/wiki/Advanced_Passenger_Train
As alavahalf explains, The purpose of the tilt mechanism is to counteract centrifugal force. Here's an explanation (which is not my work):
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As a train (or other vehicle) rounds a curve at speed, objects inside the train experience inertia. This can cause packages to slide about or seated passengers to feel squashed by the outboard armrest due to the centrifugal force, and standing passengers to lose their balance. Tilting trains are designed to counteract this discomfort.
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Although the APT was introduced into limited public service it did not last long and it was not until the "Pendolino" trains were introduced by Virgin in 2002 when Britain had its first regular tilting train service.
The IC125 was a conventional train in that it did not tilt but it included some novel features (for the time) the most notable of which was a locomotive at each end. This enabled far better acceleration. The trains also had a state-of-the-art braking system which enabled them to be brought to a halt from 125mph in the same distance (about 2,000 yards) as conventional trains could manage from 100mph. This meant existing signalling spaced for 100mph running was adequate for the 125.
Mrs NJ and I travelled on one of the first 125 services from London to Swansea in about 1977. We had a voucher from a packet of “Persil” which gave us a ticket for £1.25! In contrast to the APT's very short commercial life, IC125s (or "High Speed Trains" as they are more commonly known) still abound after more than 40 years service.