How can I prioritise network bandwidth on a per application basis?

You can use force_bind to set a priority for all sockets of an application, and then, using Linux QoS (tc command), you can assign applications to a priority band. Check the README file for an example.

http://kernel.embedromix.ro/us/

Disclaimer: I am the author.

Example:

14: Force priority (between 0 and 6 for non-root users). You can
        use 'tc' command from iproute to set-up 'prio' qdisc and to
        assign prio to queues:
        # 0. setup
        export FORCE_NET_VERBOSE=1
        export LD_PRELOAD=${LD_PRELOAD}:/usr/lib/force_bind.so
        # 1. Make sure you have a 'prio' qdisc attached to eth0, for example:
        tc qdisc add ev eth0 root handle 1: prio
        # 2. Assign applications to classed (bands):
        export FORCE_NET_PRIO=6 # interactive, band 0
        your_voip_program_here
        export FORCE_NET_PRIO=0 # best effort, band 1
        your_mail_program_here
        export FORCE_NET_PRIO=2 # bulk, band 2
        your_remote_backup_program_here
        # 3. Run tc statistics so you can see the classification:
        tc -s class show dev eth0

Of course, you may use htb or any other qdisc.