I was there 2 weeks ago with my 3 year old child who is already learning how to ride and can ride a trailing horse behind a guide in a walk. But I did not take him near the guard horse.
I certainly recommend going to witness the ceremony but I do not recommend taking the child anywhere near the horse at that time for various reasons.
1) The horse is on duty and is going to follow a certain schedule as commanded by the rider.
2) The place is really crowded, you won't even get a lot of room to get very
close to the horses.
3) For the child's safety. Going near a horse on duty to pat him/her is never recommended and mostly not even allowed in a crowded place.
This is relevant even to the single horse guard that is always there at the gate. That place is small and visitors are not allowed to go very close to the horse. Even if they were allowed, there is a significant crowd making it difficult to have any significant time with the horse.
There is a Museum (The Household Cavalry Museum) which shows you behind the scenes life of Queen's guards it will be very interesting for the child to get up close to the horses there but there is a glass separator there.
There are riding clubs and stables around the area and you could take the child there to pat or feed the horse, they do not allow a 3 year old child to ride but they don't mind a pat for the horse.