This is a pretty broad question, but I've been working with general speech datasets, so I can try give an idea of what could be done, and share some things I would probably try myself.
Normally, in adult speech recognition, what is asked that people read a certain text, but this is obviously impossible with babies. So instead, I would listen to the speech files and look for similar speech samples, which should give an idea of what could be done.
I think it is pretty difficult to do anything from a linguistical perspective. At this stage, vowel formation is happening, so running some vowel recognisers might be an interesting thing to try. I believe there are some which output the probability of which vowel was formed and then you could do a longitudinal analysis of increase in vowel formation quality (based on the recognition probabilities).
I would probably look at the spectral components of the speech, i.e take a mean spectogram at each age stage and compare. I would also try Mel Frequency Cepstral Coefficent (MFCC) representation, as that is a more perceptual representation of speech more commonly used.
What has been done
As I know, child speech is an active research area, I'm familiar that there is a research group in Lisboa working on child speech and child speech therapy, see Martins et al. : Detection of Children's Voices, but "toddler speech" is a bit more difficult. One of their member's, Alberto Abad, gave a presentation here, also from the perspective of pathological child speech and development, which seems especially difficult. I found that interesting, I recommend you to watch the beginning so that you can have an idea what speech structures even exist at that stage.
The tools that you will need to undertake the analysis really depends on the choice of your programming language (Python or R), but I hope I gave you some initial directions.