Three years ago, my uncle and his wife decided to emigrate and sell their house. We agreed (in front of witnesses) that they could put up a log cabin on my property (at their expense) whilst their paperwork is being done. The agreement was that they would live here, rent free for a period no longer that three years, after which the cabin would become ours. They would only contribute towards lights and water. The three years period has lapsed and I have approached them to start paying rent. They now deny that they ever entered into any agreement and want to leave and remove that cabin and all its fixtures from the property.
Please advise what my legal recourse is on this matter.

You don't say what country, but I suspect that it is GB, because you say whilst.

I can only answer for America, but here, if a house ( log cabin is a house) is built, it is a dwelling and becomes part of the property.

So, you need to get an attorney involved, and notify them,

You said you agreed in front of witnesses. This indicates that you have nothing in writing, which was extremely foolish. But will the witnesses back you up?

I believe a structure built on your property is yours. So here's what I would do. I would give your uncle a written statement that restates the agreement. Get signed statements from the witnesses that they agree to what you have stated. Then add that since the three years are up, they need to start paying rent or vacate. State also that any attempt to remove the structure would be met with legal action.

Three years ago, my uncle and his wife decided to emigrate and sell their house. ... They were going to move out of the country in which they were located? That's what "emigrate" means. Or perhaps they were going to move into your country (which makes more sense in relation to the rental arrangement they made with you), in which case you mean "immigrate". In either case, it is mostly immaterial.

...
You don't say what country, but I suspect that it is GB, because you say whilst.
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Perhaps Canada. While it's not impossible to build a log cabin in the UK, I'm guessing it's much more common in the US or Canada.

Where did they get the logs? If they were cut on OP's property, OP would have an even stronger case.

You don't say what country, but I suspect that it is GB, because you say whilst.
We live in South Africa and they where planning to move to New Zealand.

I can't find a specific reference, but since South Africa law is based on English Common law, I would assume that any structure built on your property becomes part of your property.

So I would do what I said in my last response.

I can't find a specific reference, but since South Africa law is based on English Common law, I would assume that any structure built on your property becomes part of your property.

So I would do what I said in my last response.

Thank you for all the help. It is much appreciated.

Keep us posted as to what happens.