I don't know many who get it aged 2. Not sure of you know how DLA works - you basically need to demonstrate that your child requires significantly more help, support and supervision in comparison to a typical child.
Here is the snag:it is totally normal for a 2 year old to e.g. not talk much, to be in nappies, to have no sense of danger and therefore requiring constant supervision, to need help with eating, wakes at night etc.
Unless you DC has also health needs which require a lot of input (e.g. you doing physio, tub fed, lots of medication etc), I think you will struggle.
Have a look at the cerebra guide - it may give you some pointers and just give it a go (appeal if need be). If you don't try, you won't get. If difficulties persist, it will get easier later to get DLA, as the gap between a delayed child and a typical one usually widens but demonstrating that a 2 year old needs more care than a typical is usually challenging.
we often waited 8-9 weeks to find out if we were awarded. Then mandatory reconsideration another 10 weeks and then appeal (which was another 3 months wait). DLA isn't a quick process esp if you go down the appeal route.
Good luck and done on doing the forms so quickly (took me 3 months on typing 30 pages down).