Either is ok. There's a subtle possible difference: "I am going to apply for a loan at HSBC bank." - this tells me where you are going to be when you apply for the loan. "I am going to apply for a loan from HSBC bank." - this tells me where the money is going to come from. But - it's very slight and possibly unintentional for someone who isn't choosing their words very carefully.

You could use either. 'From,' would be more common if talking about the name of the bank that you are getting the loan from. But you could certainly use 'at,' if you were talking about the immediate future - for example telling a friend about your plans for tomorrow.

It's not easy to choice the right one...maybe "at" is correct?