Yes, a mail forwarding company can open a business bank account. It doesn't matter what kind of services you offer - you have a legitimate business, therefore, you can set up a bank account for it.
Take a look at some comparison sites to get an idea of what kinds of deals are available - KnowYourMoney, MoneySuperMarket and MoneySavingExpert are all reputable sites. When you have found a bank account offer that you like, you can apply in person at your local branch, or you can complete an application by telephone or online.
When you are completing your application, you will be asked to provide lots of information about yourself and your business. The bank will ask for your company registration number, but they may also wish to see your certificate of incorporation and the memorandum and articles of association.
The account opening timescale varies from bank to bank, but most accounts are up and running within 3-7 business days. It is often quicker to apply in a branch because you can sign everything there and then - if you apply online or by phone, you will have to sign paperwork and post it back to the bank before the application can be finalised.