Will using my phone whilst charging make it charge slower? Just the question in the title.  If I have my phone plugged in and charging, will it take any longer to get up to 100% charge if I'm using it or will this make no difference?   (I'm using a galaxy nexus)
Yes.  The Galaxy Nexus will accept no more than 4.5W of power (0.9A @ 5V) from a charger, regardless of what the charger is capable of supplying.  

This isn't a whole lot, so doing stuff will cut down on the amount of power available to charge the battery and it's actually possible to run down the battery while you're plugged in.  Testing by a guy on reddit showed peak power draws from the battery of over 5W when using google maps, so you'd end up using the entire charger supply to power the hardware, and another 0.5W drawn from the battery.