Mechanic left oil cap off. Oil everywhere. Is this a big problem?

It's not a common error but it does happen.

Personally I would have called the dealer, had them tow the vehicle back to them and make them clean out the engine bay.

If the engine lost enough oil for the oil pressure to drop ( which would mean insufficient internal lubrication ), yes it can have severe long term effects because of the increase wear on internal parts. That is why this should have been documented with the dealer that did the oil change so if something does go wrong down the road, you have proof of dealer negligence.

Please don't drive the car till the oil level has been checked! Personally, the dealer who did the work is liable for any engine damage do to low oil level and cleaning up the mess. They should be made aware if it at least!

More than likely you can do the clean-up easily yourself. Check the oil, fill the engine with the same brand and viscosity. Put the oil cap back on tightly. Take the car to a wand type car wash, open the hood and hot soap spray down the entire engine compartment. Rinse when you're done.

Guess you know where to go for service next time! I'd write a letter to the owner of the first place that did the shoddy work. Oil all over the motor not only could have started a fire but the missing oil could have been bad enough to damage the bearings if the oil pressure warning light came on.

You are okay as long as you topped off the oil level in time and did not let it get too low. 25 miles should not have done much in the way of loss.

Be sure to write a Yelp review so we know exactly who the dealer is.

do not drive the car contact the dealer and make them fix this up