I'll address each issue one by one. I wouldn't say "After several years" is an accurate translation. I'd say it gets the gist, but we'll get to making a better translation at the end.
それから
This word means "from then on" or "afterwards". Your translation of it is fine, and as it's just a word, I won't overelaborate on it.
流{なが}れる
This does indeed mean "to flow", both in the sense of a liquid, and time. 時間の流れ is a common phrase meaning "the flow of time". 流れました is just the polite past tense of this, so your translation of "flowed" is correct, though ends up sounding awkward in the sentence in English when you piece it together, as we'll see later.
It's not unnatural, in English or Japanese, to mention the flow of time if you think about it as a fluid. Not as common as 時間が過ぎます, but still used.
数年の歳月
Without going into linguistics over what の does, it can often be used in the type of phrases that you mentioned like 数年{すうねん}の歳月{さいげつ}. This part is best translated as either "many years", or "years upon years". 数年 is literally "many years", and 歳月 means "year" in much more flowery language than what most people are used to, 年. 「〇〇年の歳月 」 [1] can be used to count years, such as:
彼は論文を書くのに 3年の歳月 を要した
He took 3 years to write his thesis
の is modifying the previous expression so that you can count years (歳月), in this case "many". You're not splitting the phrase incorrectly, but just need to realize that you're counting years in this phrase.
Overall translation
Now that we know all this, we can piece together a good translation. A more literal translation sounds awkward in English, but if you're a literal mind you could think about it instead as
Thereafter, many years flowed past.
Fine English, but it doesn't flow as well. I'd prefer a translation along the lines of:
それから数年の歳月が流れました
Thereafter, years upon years passed.
Not how I'd translate this as day-to-day language, but this is a video game, and drama's important.
[1] In case you didn't already know, 〇〇 is commonly used in Japanese (and Chinese and Korean incidentally) to indicate 何々, or some value to fill those circles, like how you might say "after x years" or even with underscores "after __ years" to show that this can take whatever value it needs to.
I assume it's like "How time flies", but Japanese has "How time flows"?
: compare en.wikipedia.org/wiki/Timestream