I think warm start problems are more common. Mine would start from cold and from hot but if stood for a half hour and restarted it would spin and spin before firing. Thermostat and temperature sender fixed that.
I wonder if the temperature sender could be the problem with yours. What you describe reminds me of when my glowplugs were failing so it sounds like they're not being switched on - either the temperature's not being registered as cold or the glow plug circuit is at fault as you've already changed the plugs.