'Exchanging' old fivers for new ones is a little more complicated than it might sound. You can't simply go into a bank, hand over the old note and be given a new one (because money-laundering rules require that all bank transactions must be traceable, however small).
You have to pay the money into your account and then, if you wish to do so, withdraw it again. However as long as you go to your own bank (or to a Post Office if they handle transactions for your bank) it shouldn't be a problem. Plenty of independent businesses (such as the burger van up the road from me) are perfectly happy to still accept old fivers, as they know that they can still pay them into their bank accounts. (So you could come and buy a couple of burgers in Stowmarket with them!)