Pity the Police didn't attend given that not only was there a collision but there were also injuries. Unfortunately, I believe that you would only have had 24 hours to report it to the Police, therefore you've gone past that now.
Nothing to stop you trying to make a claim, however it's probably doubtful that you would succeed given the fact that you're the one who drove out in front of the other car regardless of what you think its driver was doing.
Go a head and try but it doesn't sound good for you.
Sorry it's not the answer you wanted.