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Only wet roads, not rain.
MSW say:
"If chain was ridden in wet conditions, cleaning not required, but some cleaning will keep wax cleaner in SC for subsequent chains:After wet ride rinse off chain and other drivetrain components with water and wipe with rag to remove dirt and dry chain. Note: recommend re-waxing chain after no more than two consecutive wet rides to prevent oxidation. If chain is extremely dirty, clean with mineral spirits to remove majority of grime before waxing."
That's why most people have a couple of chains in rotation per bike so they can just swap the chain and rewax the other when required.
I got LBS to clean first two chains. You could also do the 'container shake' if you didn't want to shell out for a US cleaner. Slow cooker was £15 from Robert Dyas so probably the cheapest part of this equation.