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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.
Interesting. Makes you wander what would happen if the sale falls though post-exchange for some reason.