Water bills are probably the least predictable of other bills. Some areas have it as part of a consolidated billing by the county and some are private water companies.
First, as others have said, I'd let the escrow officer know. Second, I'd contact the utility company and inform them of the date that ownership is being transferred.
Sewer normally transfers with ownership and does not require any notification, but the fact that you prepaid it could be just credited to the new buyer if you're not following up on it to either have the escrow, or the utility company give you credit for that payment.
Between those two parties (escrow and the utility company), you might get the correct story for your particular property.