The words have been around as familiar (in the literal sense) terms for a long time:

"Zounds! I was never so bethump’d with words, Since I first call’d my brother’s father dad", William Shakespeare, King John, Act ii. Sc. 2.

However by their nature, they will not always be recorded in writing. It is also possible that they have gone in and out of favour rather than monotonically increasing in use - unfortunately I can't give non-anecdotal evidence for this.

Mum/mom/mam/ma/mama/mummy may have followed a different path from dad/da/pa/dada/daddy/pappy. Both are thought to be derived from a baby's early utterances, with "ma" being one of the first sounds to emerge. This, and the closer relationship with a mother mean that there is no particular reason to think that the male words are always as common as the female words.

As a follow up, why did it become 'Mom' in the American English whereas it's 'mum' in English?

Both words are something that have always kinda been around, and are also found in many other languages. For instance, Mandarin has 妈妈 (pronounced mama) and 爸爸 (pronounced baba). Mom is just truncated version of mama, and Dad evolved from dada, and papa before that.

I've studied how the english language formed, and both have been around in some form the entire time. Linguistics believe that the reason why they are found in so many other languages going so far back is because they're often the first word like sounds a baby can make, and adults have assigned meaning to it from there.

Lets put it in a graph. Data sourced from books.

Mom & dad have been on the up since the 70s. Maybe color TV (>1972) or parenting education (Spock)? The {papa, mamma} peak of 1869 is interesting too.