What kind of floor shift did you install?

As the other reply posted, you should have the 3 pin neutral safety switch in the trans.

It's probably a case of the shifter being out of adjustment.

To verify that, hold your foot on the brakes, twist the key to start/crank and move the shifter back through the gears.

It will only start when the trans is in park or neutral.
Where the shifter says it is may be a whole different thing if the adjustment is off.

It could also be that the wire is broken or the plug came off at the trans.

Do your back-up light work when in reverse?
They're run through the same trans switch (the other two wires).