This sounds somewhat similar to the issue i have had with my car for a couple years but still not bothered to fix yet as I rarely drive it.
When cold, I turn the key and nothing happens. Starts straight up if I jumper the posts on the starter motor though. Also starts straight up even when it is only remotely warm still.
I have voltage at the solenoid when I try to start but clearly not the amps to actually activate it. Not sure if it is to much resistance in the wiring, or the solenoid is sticky and requires excessive amps to work when cold.