If the Arrow light goes on and off when the relay is pulsed by the shifter button, the control circuit (and presumably, the solenoid) are OK. I say "presumably" for the solenoid because the OD does finally engage. I can't imagine anything in the relay-to-solenoid control that would delay the solenoid operation.
Here is a summary of the fairly simple OD control circuit:

OD operation involves 5 components and the wiring that connects them...

1) Fuse 11 (fused voltage supplied to items #2, #3, #4, and #5)

2) The Shifter button/switch (turns OD relay off and on)

3) The OD relay (controls Upshift Arrow light and Solenoid) (separate wires)

4) The Upshift Arrow light (ON means relay and solenoid OFF)

5) The OD solenoid (Allows OD engagement when ON, the "default" Relay state at Key On)

If the Shifter switch makes the Arrow light (#4) go on and off, then items 1, 2, and 3 are OK, and the problem is most likely the undercar wire to the Solenoid (#5) or the wire connector terminal.

Less likely would be be Solenoid itself.
Also possible (but very unlikely) would be the relay controlling the Arrow light (#4) correctly, but failing to activate the Solenoid.
An internal transmission problem is also possible.
--
Bruce Young
'93 940-NA (current) — 240s (one V8) — 140s — 122s — since '63.