YES - ENGINE WEIGHT OIL ONLY in the gearbox (gearbox is the same as the transmission, BTW), if you want your overdrive to work. A straight 30 wt nondetergent oil is best.
Unless you discussed the overdrive specifically with them, the Redline Oil Co. may not be aware that an overdrive unit was fitted to Healeys that shares oil with the transmission.
Some guys I know use 50 wt racing oil and say they have had good results with that. I think racing oil is usually nondetregent, but the 50 wt spec is a bit heavier than the factory recommended. The factory's limited recommendation could have been due to the fact that the choices we have now didn't exist then, but do beware heavy weight oils in your transmission/overdrive unit.
I would check a knowledgeable source before even trying that 50 wt oil, myself. AL Bradley