You can get an impression of how much height has been lost if you look across the SSR platforms. This photo from November 2010 wasn't taken with this purpose in mind so it's not the best quality, but it's illustrative. (click for a larger version)
Found an even older image showing just how lofty this station used to be.
My notes say this comes from the National Archives but I can’t find a link so please correct me if you know otherwise.
The essential things in life are seen not with the eyes, but with the heart. --Antoine de St. Exupery
As far as I know there was no 'deliberate' design intention to echo any NYCTA station - the actual construction here in the late 1930s was quite clever as it involved under cutting shallow ground and properties either side of the existing District line station box to create the space for the Central line tracks and platform widening. When the last refurb took place the slimming back down of the central columns (back to their 1930s profile) was deliberate as we wanted to get back every inch of platform space possible given the useage here. We also wanted to take the false ceilings out to recreate the sense of height and space but - a combination of multiple requirements for use of the ceiling and above track space and, frankly, loosing the battle, gave us this current ceiling line that everytime I used the station (almost everyday) I inwardly groaned about as boy, is it low....