Your payment over the internet will have went via the bacs system which takes 3 to 4 days, LTSB wont see that payment hit your card until then, so by paying by debit card they can confirm you have made a payment instantly and release the funds. If the £300 was to clear the balance in full then its likely you will be charged interest for being late.

Most banks will be moving on the Faster Payment System which should make your experience a thing of the past (or so they say)

Why not call them back and ask them to reconsider the late fee, if its a one-off they may refund it.

And yes most credit card companies tend to work in the same way, so im not sure if its worth the hassle of moving.

Eco