Burton stuff is pretty averagely priced for the sector. They have some spendy gear and some cheaper gear.
For bargains, go to TKMaxx or check out http://www.bargainboards.co.uk (couple of other similar sites out there too).
If you need to really keep the cost down, your MTB shell jacket will probably do the job fine. Uber-waterproofness is rarely an issue in the Alps, more often you need your kit to be breathable. If you only have a light shell, layer up under it. Base layer (merino is the bomb!), 2 fleeces and a shell will cope with almost any conditions.
Get some decent snowboard trousers as it sounds like you're a beginner in which case you'll be spending a fair bit of time sitting in the snow. At least 10,000/10,000 waterproof/breathable. Shop around and you'll get something for under £100. Get good gloves for the same reason. DaKine always make good snowboard gloves at sensible prices. Something with a removable liner is best if you don't eventually want to experience the dreaded stinky-hand.
If you're hiring a board, you'll be able to hire boots along with it in a package. Boots are definitely the first bit of kit to buy (after clothing, obviously!) but it's still a fair chunk of cash if you're just starting out. Decent boots start at about £150.
If by "cushiony-soft jacket" you mean a down jacket, then you don't need one. Too hot to ride in for the most part. If it's really cold, put an extra layer on.