modern roofs are felted. If it is well insulated, the loft will be cold enough that the snow will not melt, but after a while will form a cap on the tiles or slates, and it will stop blowing in. Next time you have your house re-roofed, they will doubtless use a modern breathable membrane which will keep rain (and snow) out but allow a little ventilation. This will cost some thousands of pounds, so give it careful thought once the winter is over, and you can have a thorough job done by someone with local recommendations.
Whatever you do, please don't have foam sprayed on the underside of the tiles. It will make it almost impossible to do any future repairs without ripping the lot off, and will trap moisture between the tiles and timbers, encouraging rot.
If there is a visible wet patch, is this an area where there is no loft insulation, or it is thin? I suppose you could put some polytarp over the contents of the loft as a temporary measure, until it dries out.
Heavy wet snow usually lies on the roof, or runs down like rain, so what you are describing mostly happens when there is very cold, powdery snow and a keen wind. So it will probbaly stop blowing in if the temperature or the wind direction changes. Unless you have holes in the roof or gables?