It can be a worry with a lot of companies and even banks going out of business, that this might happen to your lender. It may not be too much of a problem though.
Although it seems that banks go out of business a lot, they actually do not. Some banks are supported by the Bank of England and will never go out of business, so if want to be completely sure then use these. They are HSBC, Barclays and Nat West. However, they do not always have the best mortgage deals, so it can be a good idea to be more flexible.
If your lender does go out of business it will not be too complicated for you. What normally happens is that parts of the business will be sold off and it is likely that you will end up being passed to a different bank. If you had savings in the financial institution, then it could mean that you might lose some of them. However, when you have borrowed money, this will not happen. Therefore you will just have to start paying the new owner instead of the old one.
After a while you may find that they change the terms on your mortgage. They may change the fees or the interest rate. You may then decide to move elsewhere, if you have not already decided to do so. Remortgaging is usually quite simple, assuming that you have a good credit record and always make your mortgage payments. Just have a look and see whether there are some better deal on the market and move to a new lender. It normally just requires a form to be filled in and maybe a few telephone calls.
So it is not really something you should worry about. If you have a reason to be concerned about your lender, then you should be able to move your mortgage then and there unless you are tied in to a fixed rate deal. So just make sure you do not get yourself tied in, if you have this sort of concern. There are many types of mortgages and most do not tie you in. Even the ones that do tie you in, sometimes have an option of being able to leave if you pay a sum of money and so you should not worry too much. Just make sure you know the terms of your mortgage before you sign up and you should have no need to worry.