If you have a good or clean credit history and you have not over your arranged overdraft limit, you can switch your bank account. During the application process to switch to your new banking provider, they will likely ask you the value of your overdraft. Most banks will want to offer to match this.
Since an overdraft is a form of credit, a credit search will be made on you by the new bank account provider. Do keep any relevant paperwork from your current bank, such as bank statements or proof of your overdraft limit, and bring this with you when you go into your new bank account’s local branch.