How can you prevent very excessive sweating under arms even when it's cold or you're not doing anything?

The most practical advice is to use an antiperspirantrather than a deodorant. Deodorants ("body spray", "cologne") mask odors but don't prevent sweating.

Antiperspirants, available as sprays and roll-ons, prevent sweating at source by blocking the sweat glands (or causing them to shrink).

If you already use antiperspirant and it is ineffective, it may be worth changing brand as different brands work in different ways. Try several brands until you hit on one that works.

If you've tried this, then you may need a more specialist antiperspirant. There are several over-the-counter brands, and your pharmacist will be happy to advise.

After that, there are some antiperspirants that your GP/HMO can advise upon.

Failing even that (which covers 99% of excess sweating), there are some medical disorders that cause sweating. Your GP/HMO can advise on these in confidence and arrange for tests. Ultimately, it is possible to surgically sever the nerve that controls sweating, preventing it happening permanently. This is a last resort that your doctor can discuss with you after you have tried all other avenues.