Why would furnace not come on immediately?
That is normal, many furnaces are designed to avoid a condition called a "short cycle".  That's where they quit, and restart again shortly after.  

It's not quite as big a problem with furnaces, but it's a huge problem for A/C units (and heat pumps!) and can damage the unit.  If the furnace is designed to work with A/C, the anti-short cycle logic may be the same for both heat and A/C.  

Unbeknownst to you, the furnace had just quit before you walked in the door.   It waited the requisite timeout period to avoid a short-cycle, then started up normally.