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Welcome to JA.
The timing belt replacement would not keep the cylinder from having compression. If the timing belt broke then it could have bent a valve and caused the low compression condition this would also cause a misfire. So what they did wouldnt cause this if a valve is bent. It would have been best to have it checked when they had the timing belt off since they would have to take it off to check the valves anyways. They asked to check it out first since they knew it may have had a problem but you told them not to so that is where you take a gamble on whether or not that it has an internal problem. So since you declined it takes the blame off of them since they asked if they could check it out. If the timing belt breaks it is always a good idea to check the valves since it is an interference engine which mean the pistons will hit the valves if the belt breaks. Sorry if I sounded harsh. But let me know if you have any other questions. I would authorize the work if the belt broke. Thanks Daniel
Take note of the ffollowing:
1. The timing belt did not break. It was intact until removed by the dealership.
2. The belt was worn out (smooth) over some 5cm - wear and tear.
3. It is reported that the tensioner is also worn out.
Ok. If that cylinder lost compression and was missing it could cause the car not to start. To lose compression you could have worn piston rings, crack in head or block, bent valve blown head gasket. So the head would have to be pulled off to check this out. They could do cylinder leak down test to give some indication of what might be the cause before they take the head off so you could ask them to do that first. But just them wanting to replace the timing belt would not cause the compression issue. Also if the belt was loose due to the worn tensioner the engine could have jumped time and then in turn bent valves.