Why is he getting so cold? Is he wearing proper padded outdoor clothing such as ski-suit/snow suit?
This ^
A standard nursery uniform is ridiculous, and thermal layers won't cut it either. In all of mainland Europe, it's perfectly normal for children to play outside for most of the day, even in deep snow and freezing temperatures, and - shock, horror - it's actually good for them, even for children with all kinds of disabilities. The key is properly warm clothing, which it doesn't sound like your DS is in if the coldness affects his limb stiffness.
OP, you say that the nursery have some all-in-ones - these are likely to be simple waterproofs as opposed to proper warm, padded snowsuits. For the sake of your DS, invest in some PROPER winter gear, which he can also use for weekends and holidays. It will transform his little life, honestly!
I would have LOVED for my children to have the chance to take part in weekly Forest School sessions, rather than being confined inside for "wet play" (ie watching a DVD) as soon as there was a drop of rain or a snowflake. But I would never rely on nursery/school to provide suitable clothing. Children who are warm and dry generally love outdoor play, but as soon as any part of their bodies (including hands and feet) get cold and wet of course they will start to grumble and prefer to be inside - depriving them of so many positives of playing outdoors (general health, concentration, IQ even have all been found to improve through Forest School sessions).