When you pee in a freezing environment, you can see the warmth coming out of it by evaporation. So it might seem that holding the pee as long as possible actually helps you prevent from getting even colder through time.
So one could say that it would be wise to keep the pee in your bladder as long as possible in a freezing environment, since your body has higher total heat energy, thus it gets cold slower.
Are there any health risks from holding your pee too long?
You should pee.
Your body creates its own heat when healthy and fed and you will easily compensate for loss of 100-200g of body mass in all but the most extreme conditions.
By the time you are so cold or in such a place as to be at risk of losing a meaningful amount of core body heat from urinating, I suspect your body functions will have shut down and you wouldn't have needed to pee in the first place.
For what it is worth: alpinists, mountaineers, Everest climbers will pee and poo all up and down the mountain.
No. Your body temperature doesn’t drop because of peeing in and of itself.
Peeing makes you lose a bit of body mass which – all else being equal – can mean that you’ll get colder slightly faster afterwards, if your body doesn’t produce more heat than it loses.
Obviously peeing usually means that you’ll have to remove a few layers of clothing and stop moving for a minute or so. Both factors can lead to your body temperature dropping (further). But I’m not sure “holding it in” helps with this a lot since you’ll just have to pee longer (albeit less often).
In any case, the dangerous aspect which leads to hypothermia and death is that you are constantly losing heat faster than you can produce it. If you are dressed warm enough exposing your bum for a minute won’t be a problem. Your body temperature might drop ever so slightly and then go up again once you are wearing all your clothes. “Holding it in” could actually be dangerous since shivering can easily make you lose control and pee yourself. In which case you’ll be cold and wet …
Urinate when you need to (and make good use of any sheltered ground you find).
You won't lose temperature, other than what's due to removing insulation for a short time.
The more important issue is that a pressing need to go will be a distraction from everything essential that you're doing. Being comfortable in that regard is safer, as you can think more clearly when your brain doesn't have to deal with a continuous "full bladder" signal.
As long as it's warmer than -45°F, you're fine.
A single event of holding pee won't harm you. It takes several days of constant prolonged holding till you develop the condition (forgot the name) when your bladder will start sending signals it's full (even if it's not). This is reversible (even if problematic), from personal experience.
A much more imminent issue is exactly how freezing is it and what is the wind-chill? It's going to be basically impossible to find yourself at the place where peeing would be painful just due to air temperature. From experience, peeing at -40°F air (zero wind-chill) is not painful at all, merely cold. So, it probably takes another 10-15 degrees to reach that point and those places are very isolated and hard to get to.
Wind-chill can be however quite painful. Way, way, way more painful than inserting camera into your bladder at the doctor's office. I don't know if it's a cold tolerance, but I peed dozens of times at around -40°F wind-chill (it's basically omni-present for up to 8 weeks per winter here in North Dakota) and it was fine. However, once at around -50°F, the sensations and pain was quite horrendous. I could feel a quite sharp and strong pain way deep inside my body and cold spreading to other neighbouring tissue around the bladder. My bladder was still quite sensitive next morning and it was still a bit painful.
You just might prefer to choose to avoid that experience.
