Any time of year providing there is a 4 day dry weather window for your area.

Weather changes come unexpectedly fast in autumn and spring. If your weather is most stable in the summer do it then and as an added bonus you have more daylight time to do it in.

There are many new paints on the market now that only require a few hours to dry. Take your question to the nearest hardware store or home improvement center.

Most of the paint/stains used for fences need a minimum temperature of 8c. to dry properly.

Summer through to autumn - but only if it hasn't been raining most of the time. The wood will be driest after a warm summer, so doing it sometime after that is best. Wood in the garden tends to be at its wettest in spring in the UK.

OH does it in spring so the climbers can be laid off the fence [I do that or he ends up cutting them back/snapping them] so he can paint behind them.

I like using those power air guns. It goes quickly, no drips, gets into the crevices and really fun to use. Just use a drop cloth over your plants and shrubs, etc. Consider using a stain, it let's the natural beauty of the wood grain to come through.

More expense !
This weekend will be fine. Just watch those clouds coming in over the Atlantic.