Causes for hyperpyrexia-

If the patient has sepsis( severe blood infection).

If there is a sudden intracranial hemorrhage.

And can also be a result of heat stroke.

Generally, environmental cold probably wouldn’t prevent/”cure” a fever.

We humans have a thermostat in our brain that recognizes our body temperature. Normally speaking, it is set at 98.6F/37C*. The body uses a variety of mechanisms to help maintain this temperature, such as shivering to raise temperature if it gets too low, and sweating to lower it when it gets too high. While the body is generally very good at this, keeping it within a degree of normal, environmental heat and cold can, in certain circumstances, overwhelm it’s abilities, leading to hypothermia (from environmental cold) or heat stroke (from environmental heat).

When we have a fever, the brain-thermostat’s “set” temperature increases, and it understands this new value as normal. Instead of trying to maintain a temperature of 37C, it begins to maintain one that is at some new, higher value (though to actually “count” as a fever, this set-point temp must be 38C or above). No matter what the temperature outside the body, the brain is still aiming for this temperature.

If the cold was overwhelming enough to force this temperature down, you might see the fever drop, but the body would be treating that drop like hypothermia- shivering, bundling up, and trying to increase the temp back to it’s fever set point. This is one of the reasons why it’s not effective to treat a fever by putting the patient in ice water**.

So to answer your question, yes, they would still have a fever, and it likely wouldn’t change the ultimate temperature of it unless the cold was extreme enough to cause hypothermia.

As for whether the infection would “slow down or speed up”, I’m not sure exactly what that means, but stress from the low environmental temperature and maintaining a higher body temperature though it could rob the body of calories needed to fight the infection, which could reasonably result in the patient being sick for longer.

*Recent evidence has suggested it may be up to a degree lower than this