Marcel,
I've never pulled a No.52 down either as my 202 isn't a 202A, but as it was loosely based on the Garrard Super motor, I'd imagine they would have used the same type of oil which was distributed in a similar way.
Unfortunately I've never found any original data for either motor that states oil types or amounts.
As far as I'm aware, the only real difference between the two motors is that the Garrards pump is activated whenever the crank it turned to wind the springs, and the HMV's pump is manually operated.
In the Garrard motor, oil from the sump is drawn up a tube and released into a reservoir in the top casting of the motor. Narrow creeks run off the reservoir so that oil can reach small pools around each of the top bearing, and small holes are drilled so the oil can feed the bearings & drain away fairy quickly. The reservoir has a nickelled cover plate that is easily removed for oil top-ups & to prevent any foreign matter entering the motor.
When I pulled my super motor down, the springs were only lightly greased, I think because oil runs down the spring barrel pinion, allowing a small amount to enter the spring barrel, so I assume it was enough to to keep the springs wet, but not enough to wash excess grease from the barrel into the sump?
I have no idea if the same applies to your motor though.
If only one spring in your motor was greased, I assume the springs are in separate barrels, and the top spring barrel contained the grease?
That would indicate to me that the bottom spring relied only on oil for lubrication, so the oil level would have to be high enough for at least part of the bottom spring to be constantly sitting in oil. Perhaps half way up the bottom barrel might be a good level to aim for?
Personally, I think you're better to have a little too much oil than not enough, and as long as the bottom spring is getting oil, and the oil isn't high enough to interfer with the spinning of the governor, you should be pretty safe.
I too would be hesitant about using transmission fluid, partly because you don't know how it may react to the different metals & gaskets of the motor, but mostly because it's probably nothing like what was used originally

For my Garrard, I just used a full bottle of Singer sewing machine oil, as it is light enough to be drawn up by the pump easily & flow around the motor as I suspect it should, plus it was the right amount to fill the sump to the level I assumed it should be. Although transmission fluid would certain circulate in much the same way and may not cause any damage, a gramophone motor won't be exposed to heat or stress like a transmission, so I think some sort of light household oil will be sufficient.
Unfortunately, unless some sort of instructions or service data can be found, all we have to rely on is guess work
BTW: Be sure to check the rubber plunger/grommet/flange thingy in your pump too. It would have probably perished to some extent by now & will need replacing.