That sounds about right, although I'm pretty certain that in one of the more cobwebbed corners of my memory is recollection of a few bits of MoD calling the place T&EE BDN during the DGT&E period. Not, now you mention it, probably ever an official title but I think that it got used nonetheless.
On the subject, we also got RAE --> RAE over at Farnborough (Bedford, West Freugh...) when it briefly changed from Royal Aircraft Establishment to Royal Aerospace Establishment circa 1988.
What's still lost on me is why they couldn't still be called RAE and A&AEE, but both happen to be parts of QQ, just as they used to be parts of MoD(PE), and before that Min of Tech, Min of Av....
G