This will be a bit rambling but have patience...You're right mate, I know but we really can't afford to get this one wrong
Tin hat on...
The first thing to be decided is a definition of success or failure.
Obviously promotion would be success but given that you have a squad that needs both reducing (because of wages) and extending (lots of games) maybe 7 or 8th place would do as a step on the way back. The squad needs to be turned into a team and the new manager and his (hopefully new) backroom unit need time to get a grip on the clubs culture and current attitude...just walking in and saying "okay lads, now we're going to do this" probably won't work.
Whoever gets the job is going to need time and drawing comparisons to my lot up the road (or anyone else) won't be helpful. The job ahead is huge and shouldn't be underestimated. Further, the perceived unattractiveness of the NE England won't help in manager/player recruitment. The turning of the squad into a team will be a huge job and every day that you are managerless only increases the difficulties ahead and the consequent damping down of the level of ambition in defining success or failure. The less time the coaching staff have, the more they have to rely on the existing scouting network (which hasn't really covered itself in glory recently...).
You also need a manager who can change his tactics, both within a match and within a season. Once everybody has figured out how you play, they'll go old school and close down all the available space. This turns matches into attritional grinding, and one goal victories are more than good enough.
What would I do if I had just bought SAFC...?
Try to appoint Roger Schmidt, most recently of Bayer Leverkusen (GER). I'd give him a three year contract and expect him to bring in his own back room staff. He won't be cheap but he'd bring a new perspective with him. Furthermore anyone who has travelled in NW Germany will se industrial and climate similarities to NE England.
And I would do it as fast as possible...