Try /r/askscience for the technical details, but they probably won't be familiar with the history. A /r/geology would be better. In brief, this is nothing to do with tectonics - rather stratigraphy was key. Early geologists, particularly Lyell were fortunate to work in Britain. Britain has a very complex geological structure, which led them to explore the importance of strata - the order in which they overly each other, the mechanisms by which they are distorted, and the rate of deposition or erosion.

This was accompanied by the discovery of dinosaurs in Britain, by such people as Mary Anning. Because they were in geology which was already under investigation, it was quickly clear that these relics were very old, of the order of tens to hundreds of millions of years. Victorians were fascinated by geology, and by dinosaurs in particular, and geology became the first "glamour" science with outdoor lectures on geology being attended by up to 20,000 people.

Exact dates took longer to obtain. There were attempts to estimate the age of the Earth in the 19C. For instance Lord Kelvin looked at the rate of cooling of the Earth and came up with an answer of about 60M years (he didn't know about radioactive heating of the core). It was known even at the time that this answer was inconsistent with the evidence of geology, but better results had to wait for nuclear theory, and for radioisotope dating. I'm afraid I can't say when the dates for different dinosaur species were fixed, but the fifties would seem reasonable. The usual interval is given as 231-66M years BP, although the termination date is a matter of definition and there is good reason to say that present-day birds are dinosaurs.

This could be a more suited questions for /r/askscience