You should be fine with a natural tree. The only problem can be if they have been treated with a flame retardant spray or other chemicals to help keep there needles.
There are many types of Xmass tree sold in the UK, and I think the one you have has the softer needles.
If it has not been treated I would think you're OK with them having the trunk. I did the same one year. Avoid the needles, as they dry out the needles get ridged, and this can pose a threat to the chipmunks cheek pouches.
Genrally pine trees are in their natural habitat, which is why yours have taken to it so well.
Happy New Year everyone, as I'm off out now.