Mabs I actually got the DLA before he was confirmed as cerebral palsy, just labelled as globally developmentally delayed (not the same imo). As far as I know they didn't contact anyone, just used the stuff I sent in. You can claim it from the age of 3 months as long as the child being claimed for needs more attention, care etc than a child of the same age. So for example, my son has diplegic cerebral palsy, and at 16 months old cannot sit up, roll over or crawl. He doesn't use a spoon yet and really needs the help that most 6-9 month old babies have. He obviously cannot do things that children of an equivalent age can, so he qualified. The diagnosis of cerebral palsy alone may not be enough, as you have to prove he needs more help than usual, and cerebral palsy can vary so much.
I think what also helped was that I sent in a report from a child development centre stating his problems and how far behind they thought he was developmentally. I expected to receive only the middle rate but have been awarded the higher one, at least for the next year when it will be reviewed again, but I don't expect it to change.
Until the age of 5 (unless a child has a terminal illness then it is 3) you can only claim the personal care component, but if you receive the middle or higher rate of DLA then you may also be entitled to ICA (invalid care allowance), assuming you care for your child for 35 hours a week. This may be means tested though, so you would have to check it out first. If you are receiving Income Support you will also be entitled to an increase in your benefit (a disabled child allowance). DLA is not counted towards your income when you are receiving means tested benefits although you do have to declare it. Of course all these benefits depend on whether or not you are successful in your claim. You do have a right to ask them to re-examine the claim if you are not happy with the answer, and will be allowed to provide more information. If that is also turned down then you are able to go to appeal if you think they are still wrong. It does take about 10 weeks to come through though, so be prepared for a wait!
Good luck and hope this was useful.