Tid, Mid, Miserai (or Misere)
Carlin, Palm, and Paste egg day.
Reciting it as a child, gobbledegook-wise, I didn't care what it meant, I just wanted to get at those chocolate Easter eggs! I later grasped that it had something to do with the Sundays of Lent, and customs attached.
Carlin(g)s are black peas, eaten on Passion Sunday, On Palm Sunday sometimes dried palm leaves were handed to members of church congregations, and Paste eggs (possibly a corruption of Pasch) eggs were what all the kids eagerly anticipated.
As for the mysterious first line of the rhyme, there are two explanations:
'Tid' was the second Sunday in Lent, when, it seems, the Te Deum was sung/chanted in church; Mid could refer to a hymn 'Mi Deus', sung on the third Sunday of Lent; Miserai/misere might be the psalm 'Miserere Mei', sung on the fourth. But there's also a very slight possibility, because the purpose of the rhyme was to count Sundays before Easter, that Tid, Mid was a variation of an ancient Celtic-based method/ language once used in the north of England for counting sheep. Exact spelling varies with dialects of northern England, but one, two three, four, five = yan, tan, tithera, mithera, pip. Tithera, mithera could equal tid mid? I'm not confident about this, it doesn't really fit snugly. Interesting though. It has been noted that even in parts of the US the old sheep-counting method is not not unknown, possibly brought across the Atlantic by early immigrants.
The full ancient sheep-counting method went like this, with spelling variations.
(My grandmother and neighbours of her generation always pronounced "one" as "yan", by the way.)
The sheep were counted up to twenty, the shepherd then closed one finger and repeated the count until all his fingers of one hand were down = a hundred sheep. Next he would close a finger on his other hand and begin anew. So up to 500 sheep could be counted using this method.
Regarding the mysterious custom of eating black carlin peas during Lent: there's no religious significance, but the tradition is said to be linked to the civil war of 1644. Royalist Newcastle in the north-east of England was under siege from the Scots. People were dying of starvation. The story goes that, either a French ship docked in Newcastle with a cargo of Maple Peas which were distributed to the people out of charity; or that a French ship was wrecked off the coast near Newcastle and containers of peas were washed ashore, much to the relief of starving inhabitants. Either way, a custom was born! Carlin peas are soaked overnight in water, boiled well then fried in butter and served with vinegar and bread and butter. My East Yorkshire grandmother used to prepare carlins that way, each year around Easter time.