I think there are no limits for accounts that are in the same name (the one with money being tranferred to and the one with money being transferred from) but there may be from one person to another. As it is the same bank/building society there likely won't be a fee though. Best bet is to check first as access wrote.
There are stops and measures in place to prevent abuse though, specifically to stop money laundering, tax evasion etc so they may ask your purpose and or permission from the beneficiary in advance with some of the anti-terrorism regulations in place. (Not sure if it does/will apply to you (depends on the circumstances and specifics involved) but it could).
A check usually clears same if its over 10K these days but 6K would likely take you the 3-7 (variable) working days.