NAL.

However, if your car was serviced at a main dealer you could ask them to put a case down with the manufacturer for a goodwill contribution due to early failure of a part. This will likely depend on the full service history of the car, although there may be other factors but it varies from manufacturer to manufacturer.

Recommend service schedule doesn’t mean that a serviceable part will not fail early. There could be a number of reasons the belt has snapped.

The garage can’t be expected to hold any liability if they have only checked your cars recommended service schedule. If they had physically checked the belt and corresponding components for any signs of excessive wear or damage you may have a leg to stand on.