Yes, it can have an impact. As the length of your history with any lender is an important factor in your credit score, closing down your account with your previous bank can impact your rating.

It is possible to keep the two accounts open, but if this could tempt you into running up both overdrafts that is probably a very bad idea. Conversely you may decide that you are unlikely to need to make any significant extra credit in the near future and so decide to close down your old account immediately.

In general a bank is much happier to offer credit cards, loans and mortgages when you are already a customer as they already know a lot more about your financial situation than a credit score will tell them, so if you have already been approved for a new current account the effect on your credit score may not be a serious issue.