Hi,
Because the delayed engagement seems to get worse over time, perhaps as the oil warms up and thins, it's possible there is wear in the OD. However, it might be something simpler and there are a couple relatively easy things you can try.
Also it might help to clean the filter inside the OD. That and/or a change of oil may solve the problem. I think Valvoline or Castrol 20w50 "racing" oil recommended is most commonly recommended. "Racing" oil is non-detergent.
Synthetic oil in the gearbox/OD is generally discouraged (there are probably certain types that will work, but I don't want to use my gearbox/OD to experiment!).
Lack of pressure could be due to wear and would mean a rebuild. But, these old ODs are surprisingly durable. To test for pressure, you need a high range gauge to about 500 or 600 psi, with an adapter to fit into the plug in the upper righthand side of the OD. It's a bit of a specialty item, but I've seen someone selling them on eBay recently.
Incidentally, under that plug is a ball that is the ideal place to measure adjustment of the actuating lever. A dial gauge can be rigged to measure the lift of the ball, and then the amount of clearance can be set. This is more accurate than the method using the external hole on the lever itself.
And, that ball and it's seat might be another area to check regarding the slow engagement. If it's gummed up in there, it could be the cause.
Be sure when adjusting the actuating lever that the solenoid doesn't stay in the initial, full lift "on" position more than momentarily.
The A-type solenoid has three positions: "off", "on" and "hold". "On" is a strong coil (or maybe two coils) that first pulls the shaft and lever, and should only function momentarily. If the solenoid stays in the "on" mode more than for a few seconds, it will burn out and need to be replaced. It should quickly drop into the weaker "hold" mode and stay there until the OD is switch is turned off. Incorrect adjustment of the actuating lever can cause the solenoid to stay in the "on" mode too long.
I'm not sure if I'm explaining this very well. It's covered in the manual and a few other places, probably more clearly than my explanation!
It can be helpful to take off the trans cover and drive the car a bit without it, observing the operation of the solenoid.
Incidentally, there are two different A-type ODs used on TR4s. Early ones have a larger pressure accumulator piston that engaged the OD rather quickly. Later ones had a smaller piston and engaged a little less vigorously. I don't know if there are any external clues to the two different versions. May not be accurate anyway, since only the later accumulator piston seems to be available and can be retrofitted into the earlier type. The best way to tell would likely be to check the piston itself by removing the side cover (there is a special procedure to do so, consult the manual if you plan to do this yourself).
Hope this helps.
Alan