I visited Buckingham Palace last summer and had to buy my ticket about a month in advance. I was glad I didn't bring my son who was then 2 and a half. He would have been bored. I think he would still be bored at 3 and a half. From memory, I had an audio guide which I wouldn't have been able to listen to if I had my son with me.
I did take him to changing of the guard once and had to lift him up so he could see over all the people in front. He lost interest after about 5 minutes. It starts at 11am and we got there about 5 past 11 and there were big crowds.
HOWEVER, there is a very good little playground with a sandpit in St James's Park up the Buckingham Palace end. From there, you can see the guards marching back to the barracks after changing of the guard. We've been there a few times. You can then take a nice stroll through the park down to Big Ben and watch it chime. My son gets a big kick out of that!
Hope it helps and enjoy your day there