How can I relocate hookups for washer and electric dryer?
I would exploit it by getting a dryer, or front load washer, which has controls on its front rather than back. Then push the entire unit into the convenient recess. Turn the other unit 90 degrees with its back to the  water heater, and easier transfers too.  

Alternatively, they make washer+dryer combo units in the same form factor as a front load washer alone. Really.  Toss in clothes and soap, push buttons, come back 2 hours later and the clothes are clean and dry. Would free up a lot of space in that room.