Verso rough running exhaust leak

An exhaust leak can certainly trigger the check engine light and cause rough running due to incorrect air/fuel mixture readings. If the leak is before the lambda sensor, it will definitely affect its readings. Strong exhaust smell confirms the leak. Have you checked for any obvious cracks or holes in the exhaust pipe itself, or around any of the joints? Lambda sensor failure is a definite possibility here too.

I've had a quick look, and there's definitely a sooty patch around one of the joints in the exhaust pipe, which I think is the source of the leak and the smell. If the leak is causing the lambda sensor to read incorrectly, would replacing just the lambda sensor fix the problem, or would I need to fix the exhaust leak first?

Definitely fix the exhaust leak first. A new lambda sensor will only give accurate readings with a sealed exhaust system. Otherwise, you're just putting a new sensor in a faulty environment. Once the exhaust is sealed, clear the error codes and see if the check engine light returns. If it does, then you can further investigate the lambda sensor. Based on the exhaust leak and the symptoms you describe, I'd bet you need to address that first. I'd recommend a workshop.

I'll get the exhaust leak sorted out. I took it to a local workshop, and they confirmed a defective lambda sensor after fixing the exhaust for 375€. Runs like new again!