I take it you've tried HSBC then? Should be a case of filling the online application out, them ringing you to confirm some details, then the account being setup. All within a couple of weeks.
For my secondary HSBC business account I had to go into the branch with my partner for a meeting with the business guy there, no hassle, he had the account set up within an hour and we could walk away with the account numbers. I guess this depends on what your branch is like.
I do, of course, fully recommend HSBC - their online banking is brilliant too.
Moving on, have you tried going to a different branch and booking an appointment there for whichever banks aren't managing to book you in?
Try online applications with other banks if your first one was ignored. Online applications make things very quick and easy unless you have any special requirements or are looking to borrow money.
Also just to note - lots of applications at once can mean lots of credit scoring at once, these searches on your credit file all at once don't look good to banks (according to the people over at Money Saving Expert).
Edit: Oh and as Merchant UK said, if you have an appointment with the business manager and a branch to set up your account, don't forget to bring your ID etc. They should tell you what you need to bring anyway. Turning up without ID means you will have to book another appointment possibly a week or two later, or maybe even longer!