Not sure just how much "fatter" the rear fuselage would have to be to accommodate a TF30. The J57 is a straight turbojet and would need cooling airflow around it while the TF30, as a turbofan, would now need that, save perhaps around the afterburner (could be accommodated by scoops on the rear fuselage as on other aircraft). You might need to use an A-7 intake as I am not certain of how the two engines differ in mass flow requirements. Depending on when you are doing the upgrade, an AN/APG-67 radar, as used on the F-20 and on the A-50 might well work within the existing mold lines.
Maybe, though one is still going to need some structural work to fit the TF30 in.
Another option might be to use the TF41 Model 912-B32 as proposed for the F-14B at one point (maybe as part of a bigger what-if scenario where this also comes off). Reportedly, this engine would have had 17,100lb dry thrust and 27,670lb after burning thrust. given the TF41 (non-afterburning) variants were used in the USAF A-7s, there is potentially some overlap. Another twist would be to see what a RN FAA RR Spey engined TF-8A would have looked like, especially around the engine/rear fuselage. Given the TF41 was developed from the RR Spey, there may also be some commonality there.
Re the radars, either the -66 or -67 would probably work. Either would also theoretically give the F-8 the ability to use AIM-7s and later AIM-120s which could certainly be interesting. Whilst the AIM-120 didn't enter service until 1991, an updated F-8 such as theorised here would probably have entered service in the 1920s/80s and thus conceivably still be in service until the 1990s. Anyway, just a AIM-7 armed one would be interesting - maybe a pair of AIM-7s underwing and 4 AIM-9s...
Of course, if doing this, does one keep the 4 20mm Colt Mk 12 cannons or does one also follow the A-7 and go with a single M61A1 Vulcan?