Not knowing the song, I guess "la mère à Nicolas/qu’a cassé ses sabots" may be confusing because it is a rather familiar turn of phrase.

A more standard turn would be:

la mère de Nicolas, qui a cassé ses sabots

Using "à" for a possessive "de" is familiar, and even sounds a bit childish (which can be expected in a *comptine):

le jouet à mon frère

la voiture à mon pote

Qu’a cassé is a shortened, spoken form for qui a cassé.

In spoken, familiar language, you could hear eg c’est qui qu’a fait fait ça ? for qui c’est qui a fait ça ?

It is ambiguous whether Nicolas or his mother broke their clogs (personally, at the 1st reading, I rather understood it would be the mother).

It is the particular meaning that « casser » can take in various contexts that is the cause of this problem in understanding; avowedly, the context can at times be hard to distinguish clearly; there are two essential meanings; one can be called “agentive” (the subject carries out the action of breaking as an act issued from his/her own will); as regards the second meaning the subject is only accessory to the breaking; since we have the same thing in English let’s consider a sentence in that language that makes that clear; what would you say of the sentence “She broke her leg going down the stairs.”? It is the same thing in this nursery rhyme.

It seems there is another variant; I copy it below;

Scions, scions du bois,
Pour la mère, pour la mère,
Scions, scions du bois
Pour la mère Nicolas
Qui a cassé ses sabots
En mille morceaux.
Voici les morceaux!

According to this variant the tanslation is not

• “who (Nicolas’s mother) broke his hooves/clogs into a thousand pieces!”

but

• “For Mother Nicolas
  Who broke her clogs¹
  Into a thousand pieces.
  Here are the pieces!”.

“Why is this broken clog story in a song about sawing wood?” is a question I had forgotten to consider. I’ll answer it now; I wouldn’t assert with absolute certitude that it is because clogs are carved out of wooden blocks and that necessarily, to make a pair of clogs two blocks of wood are needed, but the coincidence leaves little doubt as to that being the reason.

¹_{edit after reading of answer provided by George M}

‘Sabots’ is never hooves when you’re talking about people, it’s wooden shoes/aka clogs. And @jiliagre is right that ‘la mere Nicolas’ means ‘Mrs Nicolas’, not actually anyone’s mother, just an older woman, presumably married. It’s an older turn of phrase, and not literary, but still in use. As to a thousand pieces rather than just two, it’s a common exageration.

So, keeping these points in mind.. what this boils down to is

Let's saw, let's saw some wood 
for mrs, for mrs 
Let's saw, let's saw some wood 
for mrs nicolas 
who broke her clogs 
in a thousand pieces
Here are the pieces!

Nothing the least bit mysterious about someone breaking her clogs (and they do break, for those of you who’ve never worn them, although admittedly rarely both at once) and sawing some wood to make her another.