I would second the caution over paid-for STEP courses (as mentioned by user and roisin, I am not sure how useful they are. STEP questions are designed to be non-routine and non-standard, so being shown how to tackle a range of STEP questions will not necessarily help you tackle the next one. However, working through questions yourself (with the various on-line support) will give you practice in solving unfamiliar questions.
As user says, a lot of maths students (and probably most oxbridge candidates) are not used to finding Maths difficult, and so have not developed the resilience/problem solving skills etc that they need to do STEP. The SSP foundation assignments are (supposed to) gently-ish introduce these students to hard questions and gradually ramp up the difficulty.
The advantages that (most - not all) Private schools (and some state schools) have are:
They have experience of STEP, so can point students to resources etc possibly starting in Yr 12
They have more able students, top set might all be on course for A star GCSE so they can do more stretching work with them (so hopefully these students have found maths difficult at some point before STEP).
They have more students doing Further Maths, so may be able to separate out a whole class. Usually this means that these students will have covered the whole single Maths A-level by the end of year 12 (and have done C3 and C4). This means that they can then start trying STEP I and STEP II questions (whereas other students will have to self teach C3/C4 if they want to try STEP questions before the end of Year 13).