Hi,
I used to be with First Direct who offer all the above years ago (one of the very few that did offer the 10k limits *and* current account interest and free cheques) but they became very disorganised...
Then I went to Citibank who used to be OK but now they are very clumsy and full of idiots. Their phone access is very poor and the person is usually hopeless. They also quietly cancel the overdraft facility every 3 years so if I am overdrawn at the time, all my direct debits and cheques start bouncing; apparently this is allowed by their terms.
So I am looking again... maybe some offshore private bank might do? I have enough income. Any suggestions?