@OP
For background, some years ago there was a study done but Tech Council, I believe, that showed promotions on the managerial side were indeed somewhat faster than on the T scale. However, note that to be promoted to a higher level on the managerial side, a slot has to exist and be appropriate - you can't just promote a manager to lab manager in the same role - there has to be a larger organization or higher level of responsibility (like other managers reporting to the Lab manager). That's why promotions often result from movement (filling a higher level slot in another organization).
There were some standard times for promotions, depending of course on performance.
I once promoted a T3 (hired in as such with PhD) to T4 in just 2 years due to remarkable performance (like 14 good patent applications and several technology transfers). Got a 5 in first year. And 5 the next. Well-deserved. But I knew of a T3 (hired as such) who retired as a T3. Did an ok job, but never excelled.
Assuming strong performance and results, my general rough estimate was
T1 - T2 in ~2 years
T2-T3 in ~3-4 years
T3-T4 in 4-9 years
Beyond that, it was widely varied.