Surely this must depend very much on the specific engine type, and also aircraft type? The only advice I can suggest is to read the POH, and do what it says.
Personally, I can't move off until the engine is at least a bit warm, but then I have an unusual engine. The Rotax has water-cooled cylinder heads, so it has a choke which must be pulled out for cold-starting (much the same as older cars do, except the choke on older cars stays out so you can drive off, whereas on my Europa it's spring-loaded to return to the normal position).
So, after starting (which really requires four hands - although I can just about cope with two if I don't have a pax to help me), I grip the stick between my knees, use my right hand to hold the choke out, gradually letting it in as the engine warms, and my left hand to keep the RPM at about 2000 as I push the choke in. Usually takes about 2 or 3 minutes, but I haven't had the 'plane long enough to try it on a really cold day yet. If I were to try to taxy, I'd need to use my feet, which I couldn't do without letting go of the stick!
Also, the checklist states to ensure an oil temperature of 50 degrees C before doing the power checks. So, by the time I've done the power checks, the oil temperature is usually up to at least 60 or 70 degrees before I line up on the runway.
Of course, this won't be applicable to most engines...
FFF
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