Find a vegan cake recipe or just adapt a normal recipe using dairy free alternatives.
eg. 6 oz SR Flour
6 oz Pure spread (or whatever you normally use)
6 oz Caster sugar
3 eggs
I'm lazy and just bung it all together and whizz it with a hand blender. Pour into 2 6-7" sandwich tins and bake for 20 minutes or so. Then sandwich together with jam and buttercream (made from dairy free spread, a splash of soya milk and icing sugar). You can ice this if you are feeling creative (not sure whether regalice ready to roll/ready rolled contains nuts but it is dairy free)
I sometimes do a chocolate fudge cake using a couple of tablespoons of cocoa in with the flour and make the fudge using 100g dark choc, a splash of soya milk and some icing sugar.
Alternatively, you might be able to buy a dairy free nut free chocolate cake from Sainsbury's - I think Kinnerton used to make one for them. (Kinnerton also now do a nut free, dairy free chocolate bar - black wrapper, the Safe Chocolate Company (I think) which is sold in Sainsburys)
Watch out for food colourings if you decide to do your own - some of them contain milk derivatives.