I bet a lot of people are in a similar position. Not that I really use the account now, I have a far happier time in my Barclays account. I actually have a real hate for HSBC.
Closing a credit account could affect your score, but unless your checking account has a credit product attached (e.g. an overdraft protection line of credit) that gets reported to credit reporting agencies, it won't have any impact.
You can use https://annualcreditreport.com once per year to get a copy of your report from each of the three agencies (Equifax, Experian, and TransUnion). If your HSBC account doesn't show up on any of those reports, you can be assured that closing it won't affect the scores generated from those reports.
Credit score is only one item they look at for large loan like a mortgage. Smaller loans like a car or credit cards are already easy to get.
I don't know how much impact it has, certainly less than late payments or accounts in collections, but it can have some effect.
Also, ratio of credit used to credit available is also a negative signal, which would increase when you close an account.
Those are somewhere between minor and middling factors. It's stupid we have to play the stupid credit score game, but a small decrease in interest rates can save you thousands when it comes to your mortgage.
Two things I don't bargain hunt for anymore: Insurance, and Banks.