The art of training outside in 5°C and rain:
By some Castelli Gabba or Assos fugu, Nanoflex, some neoprene shoe covers and gloves.
Put woolen soles in your shoes.
Where a thin base layer off wool (merino).
Equip your bike with mudguards/ rain guards. Use a mountain bike or a cyclocross; you do not want to travel in high speed when it’s cold. The wind is your enemy.
Focus on doing your hours on the bike with low pulse (hr sone 1), avoid sweating: unless doing intervals.
If intervals: find a place nearby, warm up, do your intervals and get home before the wind gets you cold :)
http://www.assosfactoryoutlet.com/fugu-jack-black.html
http://www.castelli-cycling.com/Products/Women/Tops/Jackets/GABBA-W-LONG-SLEEVE/p/451455714A