Apologies for not clarifying in my earlier reply - whether you have to do the TMUA/MAT/etc depends on what course you're applying for at uni, and whether you're applying for Oxford or Cambridge! I applied for Computer Science at Cambridge, and they require you to sit their own TMUA admissions test (will be replaced next year however), at Oxford it's the MAT you'd have to sit.
Both the MAT and TMUA are quite similar in that the questions are not your standard A-Level questions; you need to do a ton of practice papers for them and really familiarise yourself with the question types, as without any prep, the questions will look terribly difficult at first glance. They usually involve things like hidden quadratics disguised behind trig functions, requiring you to sketch graphs and deduce patterns, etc - they mainly test your problem-solving skills more than the usual A-Level "memorise this formula or this procedure and you're good to go".
If you're applying to do Maths at uni, you'll definitely most likely have to do STEP (if you're applying to Imperial for CS they'll likely ask you to do STEP 2 as well) - from my understanding, it's harder than both the TMUA/MAT - but it's more difficult to compare the two, as MAT/TMUA are multiple choice whereas STEP is marked similarly to A-levels: actual examiners mark your working out & solutions, and you only have to pick 6/12 questions to answer.