A further clarification to the previous clarification...
A house cheque is paid in at the drawer's branch - i.e. the branch where the account issuing the cheque is held. The beneficiary only needs to have an account with the same bank, not necessarily at the same branch. In this case you have a house cheque which will clear on the same day. Bear in mind that many banks now have what's known as clustered branches - your branch is determined by your "sort code" - the 6 digit number (3 sets of 2, hyphenated) displayed at the top of your cheques etc. So it may be that a cheque could be considered as a "house" cheque even if paid in at a different location, if the branch you pay the cheque in at is part of the same cluster as the issuer's branch.
Secondly - there are now clear agreements on clearing. The "2-4-6" rule states that, for "away" cheques you will earn interest 2 days after deposit, can withdraw funds 4 days after deposit and the cheque cannot bounce 6 days after you deposit it.
See here: http://www.bba.org.uk/content/1/c6/01/15/79/APACS_consumer_cheque_leaflet.pdf
So, logically, if someone pays your cheque in on Monday and they can earn interest on Wednesday, the funds must leave your account on the Tuesday at the latest.