You should not need a different computer for this. On these speed density trucks the computer fires the injectors in two banks rather than one at a time, so it does not care which firing order the engine uses. Firing order is set by the camshaft, and what changes is the order the plug wires go on the distributor cap.
Ford built 5.0s with two orders: 1-5-4-2-6-3-7-8 (the standard 302) and 1-3-7-2-6-5-4-8 (the HO and 351W pattern). You are right that Ford changed it: the 1987 to 1993 truck 5.0 fires 1-5-4-2-6-3-7-8, and the 1994 to 1996 truck 5.0 moved to 1-3-7-2-6-5-4-8, the same order as the 5.8. Ford's own 1994 plug wire decal for the F-Series and Bronco 5.0 reads 1-3-7-2-6-5-4-8 and it is on our Bronco page below. So if the wires are still on the cap the way the 1993 ran, the 1994 will not fire. Rewire the cap for 1-3-7-2-6-5-4-8 before you touch the computer. The only catch is a rebuilt engine with a non-original cam, so if it was rebuilt, ask the builder which cam is in it. Look for the firing order decal or a cam tag on the replacement engine, or ask whoever built it which cam is in it, then wire the cap to match.
Also confirm the distributor is in time: bring cylinder 1 up to top dead center on compression, check that the rotor points at the number 1 tower, and set base timing with the SPOUT connector unplugged. If it cranks with no spark at all, that is a separate ignition problem, not firing order.
Firing order and plug wire diagrams for the 1992 to 1996 Bronco are here:
https://www.supermotors.net/reference/ford/bronco/1992-1996/schematicsDana