Why do people's hands get cold for the first half an hour and then suddenly they are warm when they are doing exercise outdoors in winter?
Cold extremeties are a reaction of endotherms where they reduce warm blood circulation to prevent losing core body temperature, otherwise the cold blood in your hands would lower your central blood pressure from 37'C, which makes the body less efficient and at risk of catching colds.

The reaction is called vasoconstriction.
quote:
Active cutaneous vasodilation occurs via cholinergic nerve cotransmission and has been shown to include potential roles for nitric oxide, vasoactive intestinal peptide, prostaglandins, and substance P (and/or neurokinin-1 receptors). It has proven both interesting and challenging that no one substance has been identified as the sole mediator of active cutaneous vasodilation. source
The above research can give you some clues about exercise and cold hands:

When you have exercised for 30 minutes, the above quote applies, and perhaps there is also a hormone contribution from natural opiates, endorphins, and other chemicals like nitric oxide.
quote:
The endomorphin peptides, endogenous ligands for the mu-opioid receptor, and nociceptin (orphanin FQ; OFQ), an endogenous ligand for the ORL1 receptor, have substantial vasodilator activity. source
For some very good reading on the subject, search for &quot;thermoregulation vasodilation&quot;, the corresponding pages give advanced general knowledge of the biology.