Cold start problem

Sounds like a battery or starter issue. Have they been tested recently?

Yes, both OK. Engine turns well but doesn't fire.

have you checked your fuel and or "spark"?

There is no spark when cold - but why?

Coil pick-up, Something is not telling your coil to "spark". It could be the Coil pick-up, the coil itself or even the crank sensor. I would start there, but you might want to take it to a mechanic, as to just replace these parts will cost a bit, and the mechanic can test each unit.