Probably best stuff is Evostik Serious Stuff which is grab adhesive suitable for exterior use. Comes in the usual 290ml cartridge. B&Q, Screwfix etc sell it.
Another product that works well is 'No More Nails' or similar from any DIY outlet. It doesn't produce fumes like some adhesives do but I used it to bond interior wooden skirting board to the wall about 5 years ago which has worked perfectly.
a PU sealant suggested by Wizard in the fix-it club. I got mine from ebay, like he suggested for about £5. Worked great.
You need the right type of adhesive that doesn't melt your polystyrene.
I've used 'polymer wood glue' in the past with excellent results sticking wood to anything.
It comes under various trade names, Wilko's do their own brand and B&Q sell lumberjack or some such brand?
All work the same, moisture cure so prefers damp wood perhaps ideal then! slightly foaming, more so if surfaces lightly sprayed with water and grabs firmly in 5mins, fully bonded in 20mins
whatever you do dont use pu adhesive as it comes off over time leaving you with a ripple effect on the outside and a weakened sandwich structure, ive dont it myself and over time it weakens and fails,
perhaps best to use one of the other methods suggested,
if i could do my repair again i would try to leave the existing polystyrene on and dry it out rather than replace it