It’s too late to advise you on spares, tyres and other gear so I’ll skip that.
Not surprisingly everything in the Alps is bigger, steeper, rockier and rootier, add in less grip from the rain and it can be quite tricky indeed. Good news is it dries quickly but if it’s wet/raining I’d avoid the trails in the trees.
You might like to try Red route in Les Gets around Mont Chery – take mont chery lift to mont caly (excellent restaurant) then round back of hill to start point. A few technical sections but OK in wet generally. It is best ridden on a clear day for the views.
I suggest you buy an IGN map as both the routes below need a bit of navigation – as per the uk you’re riding on the marked paths a lot of the time.
Best trail we did this year was down to Samoens – search for thread on here – it starts from top of Les Gets to peak called Bourgeoise then down to Samoens – you need to organize a taxi/trailer back (was euro 60 for three of us but well worth it)
Col du Coux – a few varients but riding there via excellent lapsia restaurant on the PdS link road from above Avoriaz/Linderets then dropping back down into Avoriaz via paths is one. Alternative is hike a bike to peak for ridge ride or ride Arette Berrois down into Champery – for these you’ll need a Swiss map.
Note if it’s wet the red trails down from super morzine into Avoriaz are quite tricky (as are return trails to morzine) there are fire road options signposted PdS.
Chatel is worth riding for Green Panorama and Blue Trails if you want something straight forward, getting to/from in bad weather needs thinking about as if it’s windy the linking chairlift can be shut.