For anyone that has downloaded new animations like Oshiel's Animation Replacer or Axel's Sexy Walk, the instructions given require you to create a mod and make a new character to use the animations. But I have figured out a way of editing existing saved games to incorporate the new animations.
These are the steps I performed:
1) Download and install the
Morrowind Enhanced Editor.
2) Download an animation mod (for example, I used
Axel's Sexy Walk)
3) Extract the animation files to the Morrowind/Data Files/Meshes folder. I also opted to rename them, since Axel used cumbersome filenames. For example, mine are "Sexy.NIF", "XSexy.kf" and "XSexy.nif".
4) Backup the save game you want to edit (!).
5) Start up the Enhanced Editor and use it to open the saved game you wish to edit.
6) On the left hand list, scroll down to Non Player Characters. Click on it and a sublist will open. Navigate to NPC_player and click on it.
7) Right click on NPC_player and select "insert subrecord". Select MODL from the list that appears.
8) Scroll down a bit until you see the MODL in the left hand list (it should be after the list of various "NPCS" entries) and select it. In the text box on the right, under "ModelPath" enter the name of the animation file to use. In my case, it was "Sexy.NIF".
9) Save the file.
10) Load your save in Morrowind and play the game!

I haven't tested this with Oshiel's Animation Replacer yet, but I imagine it should work with exactly the same procedure.