Individual banks do this, cheques in my business and current accounts are available straight away.
You can ask a bank if they will do special clearance, but it's not something they really offer any more (although it really depends who you speak to on the day, and who they speak to at the clearing house).
Incidentally, cheques never really 'clear', the cash can still be taken back from your account at any point if there is a dispute, even if you have withdrawn it.
Can you ask for a small overdraft to cover the cheque?