When you let the buyer know you made a mistake and apologise, they may or may not co-operate. They might co-operate but unsurprisingly be annoyed at the inconvenience and extra cost.
How £much is the extra postage? - maybe consider giving a discount? or if quite £small, maybe consider paying it yourself? Whatever you do, if anything, could turn them into a repeat buyer or a never-again buyer.
If the sale goes ahead, if you don't already do so perhaps consider messaging to let them know when you've despatched? Yes they can see it in their eBay orders page. But human contact can 'oil the wheels' especially after any shaky start.
If the sale goes ahead, look to see if they leave feedback. Whether it's positive, neutral or negative you're allowed the option to reply. When potential buyers research a seller new to them, it can gain or lose a sale depending if and how sellers respond to any unfavourable feedback. And leave them feedback too if in the end the transaction went smoothly, which is particularly helpful if it's a buyer with a low feedback score wanting to see it increase.
I appreciate I'm probably 'teaching you to suck eggs' in which case I do apologise, but I do hope this helps you or anyone in a similar situation.
