There is a continuum in questions of correctness between what is clearly not English and what is clearly good English. In the middle there is sometimes a grey area, in which even teachers and writers cannot always agree.
At the un-English end we have: years since snow then snow first. No native speaker, even one who had never been to school, would utter those words. Further along the continuum we have: That were the first snow what we had in years. You will hear utterances like that, but they are not acceptable in any but the most informal writing.
At the clearly correct end, uttered while it us snowing, we have: This is the first time we have had snow for years.
In the middle we have:
1. That was the first time we had had snow for years.
2. Last week was the first time we had had snow for years.
3. That was the first time we had snow for years.
4. Last week was the first time we had snow for years.
5. That was the first time we have had snow for years.
6. Last week was the first time we have had snow for years.
I have no doubt at all that you will hear all these from native speakers. That does not necessarily mean that they are acceptable in the sense that educated speakers would write them. You frequently hear aint for isn’t/aren’t/hasn’t and even haven’t and, as Barb pointed out, I seen it. That you hear them does not make them correct.
#1 and #2, above, are fine. #3 and #4 can be heard commonly, but would disturb British readers who thought about their choice of words. (Actually, only a trained phonetician could say for sure whether someone had said we had or we’d had in casual conversation.)
#5 and #6 are, in my opinion, even more marginal. In both cases, if the speaker is thinking about a recent onset of snow, the utterances could be considered acceptable, especially if there are still signs of snow on the ground. However, without further context, that could be any past time, even many years ago, while ‘last week’, as Barb pointed out, is clearly recent.