Most water-based (pet-friendly) wood/fence preservers/stainers you can get from DIY stores, should be fine. Just check the tin to see what it says.
Otherwise, you might want to see if you can get pressure-treated wood as this will be weather-resistant for a couple of years; saves you trying to paint it with preserver in this weather!
Are you getting the wood from your local timber yard? It's a gazillion times cheaper than buying it from a DIY store.