If my chimney is leaking smoke into the attic space and bedrooms, does it definitely need re-lining?

Smoke is the least of your problems and fitting a twin wall flue linner is a priority. Carbon Monoxide at 20,000 parts per million produced from an open fire or wood burning stove will kill. Please do not use the fire untill resolved.

Rerendering may stop the smoke, but my advice would be to fit a twin wall stainless steel liner. If you ever had a chimney fire, I wouldnt rely on the render stopping further damage. There is tremendous heat from chimney fires, I have seen stacks split from the heat, so for peace of mind get a liner in, preferably by a Hetas registered installer.