Mother-in-law is US Citizen born in Northern Ireland, looking at dual citizenship options

Hello! My mother-in-law was born in Belfast to American parents, grew up in the United States and currently only has citizenship in the USA.

She and her son (also USA citizen) are both interested in moving overseas. Some quick googling implies that the mother-in-law could attain British and/or Irish citizenship due to her birth (she has the birth certificate). And her son believes he could then also apply for the same citizenship after his mother attains it. And then his wife could apply for citizenship after her husband gets it.

Does this chain-reaction citizenship application process really work like that? While they have both visited the UK and Ireland multiple times, neither have worked or lived there, so they would have no shot at showing any kind of residency history. They do (of course) speak English perfect, are law abiding, and would study to pass the Life in the UK test if necessary.