Thanks all for your help.
I think I finally fixed it. My solution in case it helps others...
1. I upgraded to Ubuntu 16.04 (this may or may not be necessary as I have a feeling it was other fixes that solved it).
2. Set a password for the bitcore user. I.e. after doing the
do the following
nvm install 5.0 (note not 4 as their docs recommend)
nvm use 5.0
sudo useradd -r -m bitcore
sudo usermod -aG sudo,adm bitcore
sudo passwd bitcore
<Set a password here>
Now bitcore has access to some files it needs to modify when installing.
Finally the bitcore docs were missing the line
apt-get install libzmq3-dev build-essential
and once that was added it all installed as expected.
Using node 5.0.0
Using nvm 0.31.1
Whilst I've not tried running bitcore yet, the installation had no errors and thought I'd share this post before I forget what I did.