Possibly "Survives Books" = Ding Dong Bell? (the oldest known recorded nursery rhyme)
As I was going to St Ives ->
I could mean "me" or could remain as I
"was going" sounds like a subtractor, so "me" or "I" is removed from the word
"to St Ives" - "st" could mean "start", and the next word is Ives, which begins with an I, so I'll guess the first part is to remove the I leaving "ves".
This would suggest each part of the clue is a portion of a single word, so the final solution is probably two words.
We turned ourselves about ->
Turned is likely a reversal of letters, so either "ew" or..
The word ourselves could be rewritten as "us", which would leave "su"
We found the one who rubbed his hands ->
"We found" sounds like we need to identify a new word or letter
"the one" likely means a single letter we need to identify
"who rubbed his hands" would indicate which letter.
I have no idea which letter this would be, but filling in the blanks from the rest suggest "r"
And marched him up to the top of the hill ->
"And marched" probably means to shift the letters of "him"
Either I'm moving the whole word "him", or I'm doing a rotation of "him"
"to the top of the hill" to me sounds like placing it in the center of the word, or it could indicate that I'm taking the middle of the word "him" and then rotating the letter, but by how much? rot13 = v
So my answer for the first portion is:
"Su" r "v" i "ves" = Survives
The little boy who lives by ->
"little" is probably meaning taking portion of the word, could be of "boy"
"who lives" could mean, that exists within, and then the word "by", so taking a portion of boy that is also in by = "b"
Robin Hood Robin Hood ->
As a repetition I'm going to assume it's either a double-letter or a repeated step.
Robin sounds like "robbing" or to take away something.
A hood is a type of cover or cloak, or something on the head, so it could be that it relates to the front of the word.
This could mean I'm removing the double letter "o" leaving "hd" or else I'm removing the "cover" (left and right) of the word giving me only "oo"
Killed Cock Robin ->
Killed is clearly a subtraction.
Robin again like above is likely "robbing" or again a subtraction.
Both relate to "cock" so I'm removing letters. If I remove the c's and o, I'm left with "k"
"b" "oo" "k" = book
Not sure how "Survives Book" relates... If I assume this is the "correct" method, then perhaps there's another step I need to take. Maybe the solution is yet another cryptic puzzle? If so, then the oldest surviving recorded nursery rhyme would be Ding Dong Bell.