I accidentally got my freshly removed mole wet. Will this effect the healing process?

This should be fine. I recommend Silagen + Bioshield to promote healing. Any scarring can be improved with lasers and microneedling/prp. See an expert. Best, Dr. Emer

You should be O.K.  The main reason we say not to get the area wet is to decrease the chance of infection.  A little moisture in the area is actually a good thing, it helps the healing process but it is hard to keep it aseptic.  It's easier to keep the area clean if it is dry but it may heal faster if it is moist, therefor many doctors have their own preferred method for wound care and you may get differing opinions on this topic.  Don't worry, as long as you didn't shower in a pool of e.coli or pseudomonas bacteria you should be fine! 
Best regards,
Dr. Amy