"T&EE"? Never heard of it.
I either slept through that one, or you mean "DGT&E".
So, the Boscombe titles were:
A&AEE -> A&AEE -> DGT&E -> DTEO -> DERA -> QinetiQ -> ATEC (i.e. QinetiQ+AWC)
The first two are because the Aeroplane & Armament Experimental Establishment became, for a very short while, the Aircraft & Armament Evaluation Establishment.
Probe Probare still hangs on by its teeth, mind...