I think your best bet here would be HSBC. They will provide the required currencies, credit/debit cards, and very easy to use online banking transfers. This includes an online "Global Account View" which features all of your accounts on a single screen and allows you to "drag and drop" money between accounts.
Regarding fees, I suspect you will need to be a "Premier Account" holder in order to avoid any fees imposed on transactions such as money transfers and exchanging money between currencies. In my experience HSBC offers extremely good exchange rates when exchanging "large" amounts of money ( greater than $10,000 / GBP 5,000 ). Exchanging small amounts will carry a larger spread but still much better than most banks offer. In my experience, exchanging GBP 5000 will have a spread of about 0.50-to-0.75 percent, while exchanging more than GBP10,000 will have a spread of as little as 0.10-to-0.20 percent.
In order to qualify for a "Premier Account", if my memory of HSBC UK serves me correctly, you will need to have at least GBP 50,000 net across all of your HSBC managed accounts, including stockbroking and other investment accounts.
In order to open a banking Swiss account, you will need to travel to Switzerland and apply in person. You cannot open a foreign bank account remotely. With a foreign investment account, I believe you can open accounts remotely. For example, I opened an account with Fidelity Switzerland using my Fidelity UK account directly from the UK, however obviously Fidelity does not provide banking services so this is not of interest to you.
The simplest thing to do is to visit your local HSBC branch and discuss it with them in person. Other UK banks, such as Barclays, will also provide such services, but in my experience they are not as competitive on fees.