As in England winter has been very wet. It did not snow and I think it
will not now as spring is coming very soon .
"I used present perfect because winter has not finished but I used past simple because it should have snowed before now it is too late for snowing, is past simple a good choice or should have written it has not snowed."
If winter is not quite over, as you say, your reason for choosing present perfect has been wet is good.
If it is now too late for snow, then your reason for choosing simple past it did not snow is also good. The part of the year when snow is possible is completely behind us.
But if there is still a chance that the "snowy season" is not quite finished, then you could say "it has not snowed". That choice indicates that you consider snow still a possibility.
The choice of tenses reveals our thinking when we speak.