Sold vehicle but the buyer hasn't completed the trade

Hamish. No. Unfortunately unless you had that stated as a condition on your auction you can't keep the buyers money. I'll admit I've made the threat of storage fees at $x per day on a few occasions though to tardy buyers who have paid but not collected.

About all you can actually do if they don't come through is relist the item and refund the deposit. I get that's not ideal, especially if it's something you want rid of.

Now that you have sent through an ultimatum email hopefully that kicks your buyer into action.

Callum,

According to the AA for Private Sales...

There is no ruling on deposits, but it is a general understanding that a deposit could be lost if the deal fails, unless terms of refund is negotiated beforehand.
If terms of refund had been negotiated, then you have formed an agreement and could be considered a verbal contract that the money would be returned if the sale failed due to the terms of the agreement.

Hence removing my earlier comment.

Hamish,

You are Best to confirm this with the AA or your local Citizens Advice Bureau - Cheers.