Rain won't really have much impact. You don't want water to get into the core, so I'd wait to drill the holes until you're ready to do the full repair, but once you're done injecting, rain shouldn't really be an issue. The epoxy initially cures via an exothermic reaction (releases heat as it cures), so to be safe I'd try to do the repair when you've got a few hours rain free... I doubt the rain could impact the curing, but still. The epoxy will begin to solidify in about 20 mins, and will be pretty much solid after an hour or two. The full cure takes around 10-15 hours, but after an hour or two I think you'd be perfectly fine if it got rained on.
https://www.westsystem.com/hardener-selection-guide/