I ran gdm3 under strace both with and without the network connected.
When the network is not connected (so no nfs directories are mounted)
gdm3 shows the greeter screen. When the network *is* connected, so the
nfs directories are mounted the greeter screen is not showing.
I examined the strace output, expecting to see it having problems with
my home directory, but no, there were many EACCES errors accessing
/usr/local. I'd forgotten that /usr/local was also nfs mounted.
I've removed /usr/local/ from the fstab and now the gdm greeter works
even when my home directory is nfs mounted.
At the moment I can't be sure what is the exact problem (The strace
files are insanely large, and I think they don't capture everyting),
I'll keep looking.