I've got a series 3 88 but the engine is making a knocking noise and a friend is buying the engine off me so I can get a 200tdi or a 300tdi engine and put it in and all work will be done by me.
So you can swap in a Tdi with relative ease. A 200 is easier, but you might find it easier to find a 300Tdi engine. You can retain everything else such as gearbox, axles etc. This would also enable the vehicle to keep its identity.
A Tdi can break a Series gearbox, but it'll be more how you drive than power from the engine. Same with axles. Drum brakes should also be fine with a Tdi, although there ways to upgrade to disc brakes.
I would also evaluate what you want to use the vehicle for what your annual mileage is likely to be. The Tdi is a good engine, esp in its day and it is frugal. But it is also very loud and unrefined. If you don't need the economy of the Tdi, there might be nicer engines out there to consider.
The Rover V8 is a well proven conversion and would make for a nice vehicle. Slightly more involved over a Tdi swap, but not by a huge margin and likely similar costs. I also personally think the 2.0 T-Series Rover engine, as used in some other Land Rovers would be a nice swap. Refined, smooth, more power than a Tdi and more torque than a standard 2.25 petrol.
Of course you could also look at retaining or replacing the original engine. The 2.25p is a nice engine, although under powered by modern standards. But they can be tuned to around 100-110hp for moderate money. Or you could even look at building a custom petrol 2.5 turbo utilising bits from the 19j TD engine and some creative thinking.
You may also want to consider what you want from the vehicle. Retaining the stock gearbox and axles will help keep the character of the vehicle. Now you can change these. An LT-77 or R380 5-speed from a later Land Rover can be made to fit. Along with the LT230 transfer case. But you'll be into the realm of custom gearbox mounts, custom propshafts and fabricated sheet metal for the transmission tunnel, floors and maybe seatbox. But by this stage it will feel less and less like a Series Land Rover and something like a Ninety might just be a better bet.
Defender/RRC/Disco1 axles can also be fitted. But as they are wider, it'll just make it look more like a Ninety. Some fab work to switch from coil to leaf springs on these axles though.
Other engine swaps you could consider.
Ford Essex V6. Used to be a common'ish swap and you might still get some bits for it. But I don't see why you'd do this over a Rover V8.
Td5, but likely to require more R&D and would entail a likely transmission swap.
L-Series diesel.