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When there's a short gap between exchange and completion the deposit is 'held to order' of the solicitor (i.e. the one acting for the seller) - no real point in sending it as that just duplicates the work & admin involved for solicitors and banks...
Holding to order means the other solicitor can demand it if / when they need to, and the party holding it has to send it.
If there's a longer (few weeks / months) gap between exchange & completion then the deposit normally will be sent up the chain.
My Solicitor told me (two weeks ago) that the money rarely gets passed up these days anyway, it's just held by the first solicitor until completion.