I have decking that looks in desperate need of some protector or oil. In the UK it's already quite wintery - temperatures are consistently around 5C (~40F), most days are wet. The instructions for all the protectors and oils that I see say to apply in temperatures over 10C-15C and only in dry conditions. How important is this? Should I assume that doing something is better than doing nothing? Is there any advantage to protector over oil in this situation (I think protectors tend to be water-based, so maybe less sensitive to wet weather?).
The temperature range is important because it dictates the drying time and should be followed pretty close. The dryness factor is critical. Wood absorbs water. When you set out to protect your deck you want the deck to absorb the sealer. If the deck is already wet, then the sealer won't sink into the wood and you've wasted your money. The deck must be dry so it absorbs the maximum amount of sealer. This applies to water based sealers too.
If you feel the need to do something now, You can get a water proof tarp and cover your deck and then apply the sealers when the deck is dried out. Make sure to allow for air flow while the tarp is on the deck. Then keep the tarp on the deck until the sealer dries, still allowing for air flow. Otherwise, wait until spring. Good luck.
It doubtless took years to get into this state. In winter, with cool temperatures, biological activity (mould, fungus) aren't going to proceed with any great speed. So, waiting until spring should be fine with very little additional deterioration.
Inquire with locals for confirmation: There may be merit in applying a fungicide at this time. In the horticulture trade Captan is used, as it's hard on most fungi, but benign to plants. This would stop any fungi in places you can reach.