It sounds like you are possibly missing the back drive linkage. This linkage is attached to the shifter, sometimes via a cable, to a rod that attached to an arm on the column. This is what acts as the reverse lock out, and the switching arm for the neutral safety switch.
I'm not sure about your car specifically, but in many instances, when headers are installed, the back drive linkage is in the way and must be removed. When that occurs you must jump the NSS. This will allow the car to start, but the NSS is ineffective and will allow the car to start in any gear.
This all seems plausible considering you know the car was being changed from a column to floor shift. You should be able to get the proper back drive linkage for the car from any of the numerous restoration parts dealers.
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-Jason
1969 Pontiac Firebird
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