Surely you'd base this on the XC-99 with the massive troop-transport fuselage, rather than the stock B-36? Assuming the modifications are major enough you'd need new-build airframes rather than bolting bits on to a spare bomber. You might start off designing just a flying radar but with all that space the idea of using some of it for command and control is surely going to come along pretty quick.Honestly I could see this as the Air Force "answer" to the Navy's Radar blimps and a way of NOT constructing the Texas Tower radar system. The airframe could probably actually carry a rotational search radar and a pretty hefty height-finding radar though you'd essentially have to turn one of the bomb bays into a pressurized space for all the gear. Initially it would be pure "Airborne Warning" as the idea of an airborne command and control aircraft wasn't around yet. A great deal would depend on what kind of altitude the beast could reach and what kind of range and endurance it could retain.
One thing is the Air Force is going to initially be loath to pare down the system to fit into something as 'small' as a 707 but they won't have a lot of choice at first. They might choose to take a hard look at trying to pack something into a similar sized cargo airframe instead.