if you leave the house in july, and you have the mild need to urinate, it’s fine–you can walk around for hours.
yet if you leave the house in january, and you live somewhere cold–like moscow or montreal–being outside in the cold makes you have to pee so much faster, sooner.
what’s up with that? 
In scuba training, we learned that contact with cold water causes the renal blood flow to increase leading to the “need to pee”. Thus we know there are only two kinds of scuba divers… those who pee in their wetsuits and those who lie about it. I imagine cold air does the same thing.
Activity in cold weather causes hypothermia to varying degrees. Symptoms of this are shivering and frequent urination; the more severe the hypothermia, the more severe the symptoms. This can lead to dehydration, which is why it’s important to drink as much liquid as you would in a hot climate.
As to the biological process of what causes the urination, I’ll have to leave that to a doctor.
IANADoctor, but… in cold weather, generally speaking, you’re not sweating as much. I remember a hot summer day when I was helping some friends move, and even though I drank seemingly litres of Gatorade and water, I never had to use the bathroom! All that fluid intake could only barely keep up with my sweating.
In my Wilderness First Responder courses, we learned that it is because when we enter the cold, our body increases blood flow to the “core” or most essential body parts–the brain and the organs in the torso–and away from the more expendable body parts, like the hands and feet. That increased blood flow leads to the need to pee. I don’t remember exactly how, though.
This seems like a useful response, evolutionarily.
As warm-blooded creatures, we have to expend energy in keeping our bodies near ~98.6°F. Including all the contents of that body. By urinating, we are expelling a cup or two of fluid that we would otherwise have to spend energy on keeping warm. And there’s no particular benefit to us in keeping it warm, since it’s already been processed by the kidneys and is just waste waiting to be eliminated. Eliminating it right away leaves more energy to keep the rest of our body warm, so it’s a helpful thing to do.
One of the key hormones regulating the body’s handling of water, so-called antidiuretic hormone (ADH), is inhibited by cold temperature (see here, under section “D”).
Normally, as ADH levels rise, the amount of water excreted in the urine decreases. So, inhibition of ADH, as occurs with exposure to the cold, will lead to increased water excretion into the urine. This contributes to the need to urinate more frequently if you’re out in cold weather.
But what about all that energy wasted melting snow?
A doctor who had studied this explained it to me as follows:
One of the ways your body cools itself when warm is to increase circulation to the extremities. Accordingly, as the ambient temperature decreases, your body seeks to conserve heat by restricting peripheral circulation. Part of this involves decreasing the total blood volume, and one quick and handy way to make that happen is to cause water in the bloodstream to be excreted by the kidneys into the bladder. Thus does decreasing temperature make an empty bladder full.
It also follows that as you warm up you must drink, or else be dehydrated.
T-bonham: It would seem that the heat-conserving value of getting rid of already-warm urine in your bladder is minimal - urine doesn’t have an independent way of causing your body to lose heat. It’s kinda like the suggestion that your freezer does less work if you remove a couple of potroasts.