It is true for almost every language in the whole world that the word for mother starts with a ma or mu sound. (Apart from the Georgian language, but there must be an exception)
In most of Britain Mum is commonly used, and younger children say Mummy. In Wales, Northern England and Ireland Mam is more common, and in the West Midlands it’s actually Mom.
There are some nice maps from Twitter here showing the distribution of mum/mam/mom.
The variety is partly because of accent, and something that’s not on these maps is the word I use. I’m from the West Country and we say Mur.
Grandmother also has a lot of versions including: Grandma, Gran, Granny, Nan, Nanny or Nana.