The next step after applying is that they'll contact XH and ask for details of his income. They don't care about your income. They'll also want to know how you want the money to be paid (it's your choice). If your XH is a reasonable sort then you can opt for him to pay the money direct to you. It's faster that way, just keep records and proof of payments.
If he's an arse, then I'd recommend that you ask for him to pay the CSA and the CSA then to pay it to you. That way they keep direct records of his payments. They can also take the money direct from his wages but I think they prefer to give him a chance to pay voluntarily first.
Once they've got details of his income they'll send you both letters saying how much he should pay and from what date. The clock for payments starts ticking from the date that they first contact him.
How long it takes is hard to answer and depends on how co-operative your XH is and also random chance. Sometimes the CSA works tolerably well, sometimes it's an utter shambles. I have heard some cases be organised in a few weeks. If your XH ignores their letters then it can take months. Phone the CSA at least once a week for an update (they do cock-all if you don't).
If he's self-employed then it also makes it trickier to prove his income. The best case, from your point of view, is if he's a PAYE employee as the CSA can then go to his employer's payroll department and check that he's telling the truth.
If payments stop, or you think he's earning more/less than he was, then phone the CSA. Don't expect them to notice because they won't.
Does that help?