However the alloy wheel is a different size to my current ones. Will this require a new MFK?
A car model is homologated for more than 1 wheel/tire size. If you have an 18 in wheel now, quite probably there's a 16in wheel size compatible for winter. Just search online which wheel and tire sizes are compatible with your car.
In my car there's a sticker on the driver door frame indicating the recommended air pressures for the compatible wheel/tire sizes. Lots of cars have this too, it may save you the online search.
Finally, most of people understand what 245/40R18 means. But one overlooked measurement is the offset. A wheel/tire combo of the same size you have installed today, but with different offset, may rub the mudguards or make the steering feel like crap. Some crazy people use spacers to fix offset, don't do this https://www.lesschwab.com/article/wh...el-offset.html
PS. A past thread on homologated/non-homologated wheels https://www.englishforum.ch/transpor...es-wheels.html
Tom
This assumes you do the MFK with the new rims!