This is a super awesome question.
Remember, in this question "peculiar" doesn't mean strange or odd, it means "characteristic of".
And "old english" does not mean the Anglo-Saxon language "Old English" spoken from 400 to 1100 A.D. It just means traditional English nursery rhymes, mostly written during or after the 1600's -- when most English nursery rhymes originated, though some go back to Medieval times.
Check out some classic english Nursey Rhymes, and not even considering the much different spelling, syntax, and vocabulary, there are some distinct characteristics.
The main one is that these verses were passed down orally from parent to child or even child to child. Because they were often not written down, sound-alike words and other distortions have built up over the centuries.
Another major peculiarity is in the use of "idiom" (meaning in this case phrases that differ from their literal meaning).
There are many myths about the use of idiom in english nursery rhymes, for example claiming that Ring-a-Rosy is about the Bubonic Plague. This has been widely discredited, yet a simple internet search will show zillions of pages about this.
These errors come about because catch phrases, rhyming slang, and inside jokes from hundreds of years ago can elude easy understanding.
The rhymes use "doggerel", or trivial rhymes with sing-song rhythms. This may because they were intended to instruct children.
The language and ideas are everyday and are intended for children. This is not obvious, because other cultures reserve songs and verse for sacred events and are adult only activities.
Check out the Oxford Dictionary of Nursery Rhymes for more insights into the history and linguistics of these wonderful poems.