If we take a lamb in for euthanasia, our vet will come out to the car (round the back / agricultural entrance) to do it. And no, they don't make us wait either, if we've called on ahead.
Or, in a case like this where it's time but one more hour or less, or even day or less won't make a huge difference, you could always ask the vet if they'd call in and do it as they are passing - generally they don't then charge the full callout fee.
Or the hunt or the local fallen stock collector will dispatch on farm. They often can't come straight away but will usually be there within 48 hours. For a case like Joanne's, a favourite lamb, I'd ask the local hunt. They help people with old horses and are very understanding.