Running

not a recommendation, but the nike store up the side of the kings cross is an outlet store so I'd head there for a scout if you're London based

I have a couple of these: https://www.wiggle.co.uk/dhb-long-sleeve-run-top-1/

Or get similar (and cheaper) from Decathlon.

I run hot and I'm not quite in kl dress code but most of the time I'll be in short sleeve running top, shorts, socks, shoes and a cap (to keep the sweat off face) for most of the year.

If it's cold (and wet) I may move to a long sleeve running top like the above, also may add a buff round the neck (which can be pulled up to cover the face until I've warmed up), and thin gloves (which never last more than 30 minutes before I'm hot enough to have to take them off).

I've never run with any form of a jacket or second layer on. I'd just boil.

You need to dress so you such that you're at the right temperature 5 minutes into the run. This means starting out feeling cold. Easy to state as a rule but hard to get right without trial and error. If you're toasty warm 30 seconds into a run you'll be a sweaty mess stripping off layers 10 minutes later.

Decathlon are worth a look - I’ve a couple of long sleeved medium weight tops I use in cold weather, and the stuff is cheap enough that you can buy without risk (7.99 as opposed to 79.99)

Think Phil posted a page up on this - you tend to be much warmer and out for shorter times when running compared to cycling, so different rules apply. No need for layers, little need for insulation or waterproofness. I find gloves useful - again decathlon do some very light running gloves for a couple of quid that work at freezing temps

shorts (or tights), singlet or tshirt, arm sleeves, gloves and ear wamer/headband/hat - this is my go to at the moment.