From the data reported from the Corpus of Historical American, scientist started to be used more frequently than natural philosopher between 1860 and 1869.
The CoHA finds 538 phrases containing scientist dated 1960-1969, and 589 phrases dated 2000-2009.
Opposition continued into the late 1800s and early 1900s, but according to this Google Labs ngram, the crossing point appears to be a little after 1870:
I present a different picture:
here's a more granular view - "natural philosopher" rallied but ultimately gave way to "scientist" before June of 1874
Unfortunately the answers using the Google Ngram data are basing their conclusions on a fallacy. "Scientist" is a 1gram and "natural philosopher" is a 2gram. There are many more 2grams than 1grams, hence the frequencies are not comparable.
Edit: And now I'm not so sure.
Let me introduce two new terms "type" and "token".
By "type" I mean a unique ngram. So, in the text 'Smith, where Jones had had "had", had had "had had". "Had had" had had the examiner's approval.' there are these 7 different 1grams (excluding punctuation and ignoring case): smith, where, jones, had, the, examiner's, approval.
By "token" I mean an ngram which doesn't have to be unique, so the above text just has 17 1grams.
Clearly there are a lot more 2gram types than 1gram types. However the frequency counts reflect multiple usage and so use token counts not type counts. If in the above we count tokens rather than types how many 2grams are there?
Well, each word apart from the last word, "approval", is the first word of a 2gram. Hence there are 16 2grams in the text. However, if the text is an extract from a much bigger piece of text then "approval" will also be the first word of a 2gram if ngrams, where n > 1, are allowed to span sentences. Hence counting tokens there are the same number of 2grams as 1grams and hence we can indeed compare ngram frequencies where n can vary.