You will most likely need a word processing software compatible with Microsoft's 'Word 97/2000/XP' (commonly known as '.doc') format for writing up TMAs, at least in many computing courses. In a couple of courses, a spreadsheet software is required as well, so I suggest getting one too.
The OU suggests the StarOffice office package.
I never worked with it and never felt the need to install it. It seems to be available for Linux (at least), but I still prefer and recommend OpenOffice (which is derived from it). You'll be fine with that.
Advanced users, especially those doing science degrees might prefer to use LaTeX.