I will restrict my answer to my known field which is Germany. I am sure there are countless nursery rhymes over countless cultures and times that could be discussed.

What is more likely is that they were not masked, but that the rhymes were often co-opted by people with political leanings. In Germany it was very common, especially with the German Pub Culture, to know many songs. A typical German could easily spout off over a hundred different tunes as late as the 1800's. This fact was made easier due to the re-using of tunes over and over with new words as a new message was needed. For example, what was originally a song about drinking in a pub could easily be converted to the Star Spangled Banner.

Similarly, children's songs and nursery rhymes could be used to easily remember political tunes to be sung at a rally or to communicate a message. The words to the songs could be distributed through your church, local calendars, books, or at a festival. Various reforms and rules tried to curb all of these distribution methods over time and generally failed.

As for coded phrases, I have not heard of that outside a few specific circumstances and would not call it common by any means.

Source: James Brophy. Popular Culture and the Public Sphere in the Rhineland, 1800-1850.

Ring around the rosie was a rhyme about the black plague.