Kat, the coral to red coral bells are the Heuchera that V. Troyan is talking about. They belong to the Saxifrage Family.
Heuchera is North American. The species first called coral bells, H. sanguinea, was first found in Mexico, but it is a hardy perennial and may be grown fairly far north. It was introduced to Europe in the 17th c. It has been hybridized with other species, and is found in gardens in southern Alberta, perennial in sheltered spots.
A natural hybrid, Heucherella tierelloides, resembles sanguinea and also is called 'coral bells.'
White coral bells belong to the species H. pubescens or one of the hybrids.

Lily of the Valley is a quite different plant, also a hardy perennial. Convallaria majalis as noted above, is white and belongs to the Liliaceae Family. It was introduced to America from Europe. It tends to 'travel,' and often pops up where it is not wanted, squeezing out more desirable plants (My wife loves it, I hate it). Another species, also from Europe, C. rosea, is pink.

The tune is one of those simple things that are almost impossible to trace back; it is so similar to the melodies of other simple songs. The lyrics, of course, are modern.