I always assumed that although the location is wrong by about a mile, it was based on 54 Broadway, the secret headquarters of the SIS during WW2. I've been to meetings there when the London Underground were using it as overflow office space in the 90's, and it feels right for it. Has secret entrances and so on.
Edited to add my (quite tenuous) reasoning.
In The Atrocity Archives, not long after Mo is recruited, Bob takes her for a drink at Wagamamas, and explains some laundry history. 'Back in the second world war, they were based in a Chinese Laundry in Soho, I think. They got Dansey House when the Dustbin's new skyscraper was commissioned'. It seems that the Laundry have had Dansey house for many decades, so it's the prior property of an organisation that moved into a skyscraper in the 50's, 60's or at a push 70's.
I don't know who is nicknamed 'The Dustbin' , but it's clear from other references that it's another intelligence or related organisation, because they have training courses and liaison meetings there.
The only UK intelligence agency I can think of who moved into a building that might have been described as a skyscraper in that time is MI6, who moved into Century house in about '64. Century house isn't a skyscraper, but at 22 floors it does tower over it's low-rise surroundings. It might have been called one once. MI6 vacated 54 Broadway when they moved into century house.
All pretty thin when I think about it. I don't know who 'The Dustbin' is, although it may refer to the quote about 'Whitehall's dustbin of embarrassing discards'
And it's all just post hoc justification. I found out that 54 Broadway, a building I had worked in, was a former spy headquarters not long after reading The Atrocity Archives and somehow it rang bells.