I'd lube the chain with a heavier oil-based lube before you go. Expect it to look black and gunky by the end of the ride, but it will still be there.
I tried wax based lubes on my commuter when I lived in Vancouver. I found the lube could be totally washed out and the chain squeaking within an hour on a really wet day.
Clip-on fenders will make your rain rides much more comfortable. Salmon Koolstops do give you better braking in the wet.