I haven't done this myself ( only fitted V8 diffs to Finnys and seen others do it) but the way to fit the later 4 link under a W111 is easy. What must be remembered is that the W111 pan is very flimsy at the rear ,it's a crumple zone.

Start by lifting your car and removing the fuel tank and old diff. Removing diff is time consuming to say the least. You need to get the car on stands and stable about 3 feet off the floor.
start by disconnecting the front arms ( one large nut and three small bolts ) with a jack under the arm to take the spring pressure. Lower the jack once you have arms disconnected. Undo the hand brake cables by loosening the lever on under the tunnel so the cables are slack then undo the nut on each cable housing where it is connected to the floor panel.
Then just wriggle the cables free from the lever link.
On the dif head you will see a cross movement rod which prevents sideways movement. In the diff end is a large bolt,undo that then the smaller bolt next to it. The rod can then be pushed clear of the diff head.
get into the trunk and undo the shock absorber nuts. The axles will drop down each side so be careful when then they release.
In the center of the basck of the trunk is a plastic cover,remove this and underneath you find a rubber mount with Four bolts and a large 22mm bolt in the center ,loosen the large bolt but don't undo it yet.

Under the car now.
Undo the shock bolts on each side and remove the shock absorbers. You can now see the brake hoses ,cut them off . cut only the rubbers .
Now place a trolly jack under the diff to take the load. it's an arkward thing so make sure it balanced ok.
Now undo the large bolt in the trunk so that it's about 6 threads still engaged.Lower the diff by releasing the jack. If it won't drop,belt the bolt with a heavy hammer to loosen it in the rubber mount.
once it moves down you can remove the bolt and drop the diff completely.

Now... under the car you have to cut off the center rod bracket,the large thing hanging down in the center of the floor.

Carefully remove the brake lines and the brass mounts. you will need those later.

To replace the diff ,you will need a complete diff assembly from a W114,114,107,116 ,123 or 126 . plenty of variety to choose from and all are suitable. The ones to avoid are 450SEL 6.9's because the ratio is way too high for your car and earlier type without the antilift links. it doesn't matter if they are from a vehicle with self leveling because you can't use the shocks. try and get one with the torsion bar. I note a lot of US market cars don't have this.
you will also need the brake booster and master cylinder from your donor vehicle.
with your diff assembly on the floor,you can see where you will need to fabricate the front mount for the diff carrier and the center mount for the diff head.

Under your car you need to weld in a crossmember to run from side to side in the approximate area of the shock absorber mounts. This because the floor pan in a W111 doesn't have any strength in this area and you will mounting the diff head support there.
Once you have this in place it's then a matter of raising the diff and determining the spring mount points. You need heavier springs because of the leverage imposed by the later trailing arms is greater than the W111 arms and W111 arms being 'softer' cause the car to squat under aceleration.

With the car supported by the diff you will now need to connect the brskes using the origial rear feed line (replace it if rusty etc ) and the new hoses you bought .the brass mounts will have to moved out wards to ensure the hoses reach and wil not be stretched.
The later booster and master cylinder must be used because of the brake swap. In the front of the car is a bias valve in the brake line feeding to the rear,remove this and fit a W108 fitting in the line that asllows the two line ends to be joined once the bias valve is removed. the older style W111 master cylinder won't work rear discs.
If you have front drums or the small W111 discs you need to swap in W108 or 109 spindles and discs as a complete assembly. The W11 discs have a different size disc and smaller pitch circle on the caliper mount . This small disc is incompatible with rear disc brakes . W112's have an entirely different 4 wheel disc brake set up and should not be confused as to compatibilty.
The hand brake is a problem and new cables have to be either made or W126 SEL cables sourced so they reach the link under the floor panel.
The drive shaft is your next project as the later diffs have a flex disc mount the diff head. your only option is to cut and shut a later shaft coupling to the W111 shaft.
once on the road the car will be transformed ,with an unbelievable ride and modern handling.