Sounds like the thermostat to me. As to normal warm up, if the "T" stat has failed open, your car will eventually warm up due to engine heat climbing, the "T" stat just restricts coolant flow to help warm the engine quicker , and as the engine temp climbs, the stat begins to open until eventually it is fully open (condition your car seems to have). Good thing is if it is the stat she has failed open and not closed.