Are you sure you got a J-type? A J-type adapter is more of a housing than a plate; and TR6 used either A-type or J-type. The A-type conversion is much easier. Here's a shot from the Buckeye Triumphs site, with the adapter housing on the left and a J-type on the right
Assuming you do have a J-type, here is a partial list (just the major things that come to mind, I'm sure others will supply the ones I've skipped or overlooked):
Find an OD mainshaft and matching gear set. If I recall correctly, there was no factory mainshaft that would work with the J-type OD & the TR4 gearset, so you'll have to find a late TR6 gearset and either install it in your TR4 housing or just use a late TR6 housing. If you do use the late TR6 housing, there are some minor adjustments to make it fit (like longer studs in the back of the engine block and a shorter pushrod for the clutch slave). Another option might be to have Overdrive Repair Services in the UK make you a custom mainshaft.
https://www.overdrive-repairs.co.uk/
Might as well replace at least the bearings, seals, synchro rings and countershaft thrust washers, since you'll be doing a full gearbox tear-down to install the mainshaft (and gearset).
Fabricate a new rear crossmember to mate with the J-type. The J-type sits farther back than the A-type, so it passes right through where the original TR3 rear crossmember goes. Some cutting of the frame and floorboards may also be necessary.
Drill and tap the top cover for the isolator switch. The required thread is an unusual metric pitch, not found in most sets. Here again, might as well change the seals while it's apart.
The speedometer drive ratio will be wrong, so you'll need to either change the gears or get a ratio adapter made. Most likely you'll also need an angle drive to hook up the speedo cable.
Of course you'll need to wire the whole mess up. A relay is not necessary (although you can use one if you want to); but I would suggest a snubber diode across the solenoid to keep the inductive kickback from eroding the contacts in the relay or switches.
I think that's all the major items.
FWIW, I opted to install a J-type in my early Stag, and I'm rather sorry I did. I got really spoiled with the instant shifts and 2nd OD with the A-type in my TR3A. Much "sportier" than the J-type, IMO. The A-type seems to practically read my mind and shift before the switch has stopped moving, while the J-type takes a count of 3 or so (longer at low road speeds) to shift into OD. Makes it totally useless on twisty mountain roads IMO, while the A-type is a definite benefit.