Hi Sam,
It could be a slow solenoid, but I think more likely that it's the pressure within OD is low and that will make it sluggish to respond.
It's an A-type, isn't it? Just from memory, I seem to recall the A-type runs somewhere close to 500 psi. When the car is driven, the OD quickly develops and maintains a pressurized supply of oil internally, ready for the moment the OD is engaged. If that pressure is low for some reason, the response will be slow.
J-type are inherently a bit slower to respond due to lower pressure operation, maybe 300 psi if I redall correctly, and the fact that it doesn't use any sort of pre-pressurizing/accumulation of the pressure, the way A-type do. However, even J-type are usually not as slow as you describe. Early/TR3-era A-type should pretty much "bang" into OD. Later A-type, perhaps from mid-TR4 production and on, have a very slight delay and not quite so harsh an engagement. They operate the same way, but have a smaller pressure accumulator.
The check valve that's actuated by the solenoid could also be dirty or partially clogged, causing delayed engagement. It's under a plug on the RH side, where solenoid adjustment can be most accurately checked with a dial micrometer measuring the lifting of the check ball, and where the OD is producing can also be measured with the right gauge and connector.
A third possibility is that the clutches in the OD are slipping. This might happen if they are severely worn, or if the wrong type of oil were used.
Personally I'd avoid slipperier synthetic oils in an OD gearbox, just because of the oil-bathed clutch. I don't know of any specific problems with the Laycock OD clutches and modern synthetic oils, but I have seen them ruin an oil-bathed motorcycle clutch set in a matter of minutes.
Also, I hope you didn't use standard motor oil in the geabox. All these oils have detergents in them (some have a lot!). This tends to foam up and can introduce air bubbles into the OD hydraulics that might cause similar symptoms to what you are seeing. IMHO, it's best to use non-detergent mineral based "racing" motor oil. (Foaming detergent oils can also effect lubrication in the gearbox, and cause early bearing failures or other lubrication related problems.)
If the OD has some miles on it, i.e. hasn't been rebuilt in recent history, it might help to increase oil viscosity (sorta depends upon what you already using). John Esposito at Quantum Mechanics recommends non-detergent 30w be used in a freshly rebuilt unit, but I suspect 40w might be needed in an OD that's got a lot of miles on it and has some wear inside. Triumph specified using 90w gear oil in the cars originally (which is similar viscosity to 50w motor oil), but that seems too heavy to me. It's best to accurately test the pressure being produced with whatever oil you choose, to see if it's too high or too low after the car is nicely warmed up on the road.
If you drain the oil, check it closely for any kind of swarf, filings or chunks of metal from either the gearbox or the OD. Brassiness might indicate rapid synchro wear. Steel grey swarf might mean other wear, such as on a main shaft. If any bushings or bearings are failing and shedding a lot of stuff into the oil, this will contaminate both units but is probably actually worse for the OD which has a pump and a couple pressure building pistons, as well as an accumulator to maintain pressure, which is fitted with a piston that's got compression rings in a bore, similar to those in the engine (just smaller).
I hope these ideas give you some leads to solving the problem. Let us know if you find anything.
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