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I'm looking at upgrading my car and have narrowed the choice down to 3 possibilities off trade me. Cars have mileage in the range 131,000 - 181,000, and price range $4,900 - $5,900. Two are 2001 model cars - a Nissan Primera wagon and a Peugeot 206 XT hatchback. The third one is a 1996 BMW 328i sedan. All are beautiful looking cars in good condition inside and out. I've been told the Primera and BMW have chain belts so wont need their cambelts replaced. The Peugeot had its cam belt replaced at 126kms.

I'm having trouble deciding and wondered if you could help. Which one would be better value for money to purchase and in terms of safety and maintenance costs in the future?

These three are all quite different in their appeal and type.
You are correct in that the BMW and Nissan are chain driven, while the Peugeot is belt driven.
Of the three, the Peugeot 206 is the smallest and most fuel-efficient. However, if the 206 is an auto, the transmission shifts on this model were not to everyone's liking (prone to being a little harsh and have been described as having a mind of their own!). Otherwise, the 206 was quite a good model.
The Primera is the most spacious (being a wagon), but if it's an auto with the CVT transmission, this transmission is known to have reliability concerns and be very expensive to repair.
If the BMW is NZ New with the 5 speed auto, it was a good model but known for cooling system problems. If you decide on the BMW, be prepared to spend a little on replacing some of the plastic fittings in the cooling system, and it should prove reliable. However, it will be the least fuel efficient of the three.
Ultimately, if you are looking for low running cost and reliability in this price range, maybe a mainstream Japanese hatch like a Mazda Familia, Toyota Corolla, Honda Civic etc may offer what you are looking for.

Many thanks for this information. I've just had a AA inspection done on the BMW and its come back with a Grade 4 rating (Extreme risk of requiring unplanned maintenance and repair) under General Conditions. It also said the Power Assist had hoses leaking and pump seepage, as well as leaking oil from the engine. Are these potentially serious and costly?

Grade 4 says it all really in relation to a pre-purchase inspection. The gradings start at 1 for a really top car down to 4 for a high risk car.
Leaks from the power steering hoses and pump will be WOF failures, and while the hoses could possibly be repaired/replaced, there is still a fair cost attached. The pump is another matter and could potentially be an expensive repair, as could engine oil leaks, depending on where the leaks are coming from.
This is not a car I'd recommend.
As much as I can see from the list of cars you're considering that you prefer "interesting" cars, I'm sorry to say that perhaps a more "ordinary" car might offer the best reliability in this price range.