To do that today, you'd probably connect with the serial port and stick to text-mode use of the Pi. Then your Beeb is a dumb terminal, or a slightly smart terminal. You should be able to edit files, browse the web, run makefiles and develop software, just like it's the mid-80s.
PiTubeDirect isn't much help here, as the Pi in that case isn't running a full Linux, it's running well-optimised bare metal code and has to be dedicated to servicing accesses made by the Beeb or Master host machine.
Although, RobC has done some
impressive things shipping sound and video from the Pi back to the host. This technique allows general C code running on the Pi - but not under Linux, or any other OS, and so no access to networking.
For the (near or mid) future,
there's a very interesting project by revaldinho here which offers two FIFOs to connect an 8 bit host to a Pi (or indeed to something else) - the FIFOs and other logic mean that the Pi doesn't have such hard deadlines and can run a full Linux. There are
many possibilities open, which
are being thought about or worked on.