If its a bit wet, don't bother with a water proof unless its chucking it down - you will just sweat inside and get just as wet.
Get some thin arm warmers for the morning, and put them in your back pocket for the return home.
Cycle harder - seriously, if you're feeling cold, push harder.
Get a smaller jacket - I have the Aldi equivalent to a Montane Featherlite, its the size of a tennis ball packed, and lives in my saddle bag. They won't keep you totally dry but are slightly wind proof enough to take the chill off - the above has a wind proof limit of something like 14mph, but that is 14 less than you currently have, also stops you sweating too much. I have used this top down to a -17c start with a base layer and aldi winter top(not the soft shell pap) without any issues
You will feel colder when you start, don't dress for this - dress for how you will feel in a few miles.
Certain parts of the body warm up at different points - legs obviously are one hot area, hence why you will see a bunch of us around here cycling in shorts in winter (another option here is to get leg warmers, that are small and can be put in your pocket on a return journey). So I might be out in shorts, but be wearing wolly gloves and (light)overshoes
Saying that, Im actually wearing a long sleeve atm that I push the arms up on.
No, no, no and thrice no!!!! Arm warmers are just a way of getting cyclists to part with more money. They serve no purpose. If it's cold, wear a long sleeve top, if it's not, don't! UK temperatures don't vary that much eg today starting at 11C and rising to 19C, short sleeves.
A dim and misty 9C rising to a 20C+ in the direct sun is quite a difference.
Arm warmers can be had for cheap (£6.50 for elasticated/rubberised grippy ones, £1 for cotton).