There’s lots and lots of different systems: https://en.wikipedia.org/wiki/Yan_tan_tethera#Systems_by_reg.... Some have methera, some have pethera, many have something else.

It's not, not really. Consider where you pronounce 'm' and 'p': they're both bilabial consonants. It's pretty typical in Celtic languages to have initial 'consonant mutations' where a sound in a preceding word has got lost over time, but it affected the pronunciation of the word that followed it in a predictable fashion, which lead it being grammaticalised.

An example from Irish would be 'capall', which means 'horse', and 'na gcapall', meaning 'of the horses', as the genitive plural of 'capall' is also 'capall', but when you use the plural definite article with it, the pronunciation of the first letter changes from a /k/ to a /g/, and Irish marks this by preceding the 'c' with a 'g' (so you can see the underlying word).

Welsh and other members of the Brythonic family of Celtic languages have something similar, which can lead to 'p' and 'b' changing into 'mh' (an /m/ sound, but unvoiced) and 'm' respectively. This is called 'nasal mutation'.

So, what you're looking at is a possible remnant of nasal mutation in Cumbric.