Where do I start to look for an automatic off feature on a circuit? (Guidance)
While it is POSSIBLE that there is identifiable auto-off componentry, it is more likely that it is implemented as a software timer and that it has no identifiable individual components associated with this function.

A solution that has a reasonably good chance of working is to occasionally disconnect the power supply for a brief period. Depending how the toy "wakes up" this may be almost invisible or unacceptable - if it sings, howls, whistles or plays "The Star Spangled Banner" on waking then this is not your solution.
 This feature could be implemented by placing a transistor in a supply lead and operating a timer that occasionally turns the transistor off.
Details on how to do this can be supplied if this is an acceptable solution. 

You need to ENSURE that your mains based solution is close to "intrinsically safe' - that at a minium it can not provide enough energy at the toy to cause a fire no matter what. You know that - but it's worth reinforcing.

There are many many 555 based astable timer circuits on web.

Here is a typical tutorial page. See the astable circuit in fig 10. - thus:

Make R1 large and R2 very small. R1 then controls charging and R2 discharging.


https://www.jameco.com/Jameco/workshop/TechTip/555-timer-tutorial-fig11.jpg  

The circuit can easily be arranged to drive a transistor - ask if of interest. 

A digital timer solution would more easily implement times in the 10+ minute range at equal or lower cost and complexity.

And here are images with links to pages for a zillion more. 

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While it's off the original topic, and while I'm not expert in ANY field of child raising (except by having helped raise two children long ago :-) ) I suspect that it may be preferable to provide a means of easily mechanically re-awakening the toy/light that leaving it on. Fear of the dark and use of night lights are normal, but learning that "dark is good" is also desirable.
Also, various "experts" suggest that deeper darkness is better than sleeping in a lit environment - I don't know the merits of this but note it 'for completeness'.