In the past i've acted as a kind of support worker to men who have left the family home and want custody of their kids.
Been a couple of years, so i cant tell you what the law is, i'm not a legal advisor. But i can tell you what i have noticed.
The judge wont automatically think the child should be with the mother. It does sometimes look like that, but its really a case of The child is best off staying where he is, and that is often with mum from the outset. Unless you can prove otherwise.
The judge is not interested in who wants what, he's just interested in what's best for the child's welfare.especially when the child is too young to have a say in the matter.
If you want custody of your child, you need to show the judge VERY clear reasons. No waffle, no arguing the toss, just very black and white "I can do a better job of bringing up the boy because........."
My advice is keep a concise diary of ALL contact, attempts at contact etc. And be honest. If you told your ex you'd be round at 10am to collect your son and you turned up at 10.30am say so. Explain why.
If your ex said you could have your boy for a week, then when you asked she said no, you need to be clear if she was being unreasonable, and it wasnt a case of you springing your offer on her at the last minute because it happened to be convenient to you.
Most importantly, get yourself a friend who can be supportive every step of the way. You CAN do it without a solicitor, but if you are unsure of the legal system and dont know your way around legal documents and the courthouse, do get a solicitor. If you are on a low income, ask about legal aid.
In custody cases, all the judge is interested in seeing is who can bring the child up best, with the child's welfare being paramount.
Have you thought it through?
How will you provide for your son financially? Will you keep working and leave him in the care of childminders all day, every day? Is his mum currently a stay-at-home-mum?
Can you prove that you are capable of coping when he gets sick etc?
Sorry if that all sounds a bit harsh or negative, its not meant to be.
I wish you success, it is obvious that you really love him and want a chance of being a full-time Dad, and I hope things work out for you.

Honestly, the courts *don't* take the attitude "the child's best off with the mum". Its just that because very often the child starts out with mum, its the dad that ends up being the one to 'fight for custody' - as opposed to "mum's already got custody".
And Stu, you didnt need to answer my questions here, it was just some stuff to ponder.
Dont forget, keep a running record of EVERYTHING, dates, times, who said what etc. And get yourself a Court Buddy!
Best of luck!! x