Thanks to everyone who got in touch to help. In the spirit of helping others, here is the solution, step by step:
Before you get going, make sure your Synology DSM is up to date. Then go into the DSM Control Panel and enable NFS in File Services. Also in the Control Panel, enable ssh in the Terminal & SNMP control panel.
Step 1: ssh into your NAS as root. Note the root password is the same as your Admin password, but this will not work if you are logged in as Admin – you have to be root for the next step to work.
Step 2: edit a file called /etc/exports. It may or may not exist already – in my case it didn’t. /etc/exports tells your NAS which directories may be mounted by NFS clients (like Volumio), who the clients are and what permissions they have. My /etc/exports only has the one line so far:
"volume1/Akasaka Rokku/iTunes/Music" 192.168.0.*(rw)
The first part is the directory you want to make mountable on other devices. You need to use quotes if your directory path contains spaces, or your NAS will get confused when exporting the directory. Note also that what looks like the top level directory in your file browser on the NAS isn’t really the top level. What you see as top level in the file browser usually sits inside a directory called /volumes or /volume1 or something like that. Note also that everything is CaSe senSITive.
The second part is the address and permissions of the device(s) you want to export the directory to. In my case I’ve put the * so that all the 192.168.0.something addresses can mount the directory. The (rw) at the end tells your NAS it’s OK for clients that mount the directory to read from and write to it. You could specify a host name instead of an IP of course, and if you’re extremely security conscious, this is probably how you’ll want to do it, but I fess up to a certain amount of laziness here…
After editing and saving /etc/exports enter the command
to make your listed directories visible to NFS clients.
Final step: go to Volumio to specify and mount your music directory. Your MPD directory will take a while to update, so go away and make coffee, or do what I did – watch it for a few minutes, get bored and go off to work.
I hope some of you find this useful.