Most of your questions should be answered with a visit to the ais website.
newcastle airport
Curses! You'll have to register to login (it's all free and publicly available so don't be shy). I think you should be able to go straight to the airfield info after login though.
Hopefully you should now be able to access the published instrument approach plates to each of the runway directions. I appreciate you are not necessarily a pilot, but I think the plates are fairly self-explanatory and you should be able to figure out where thay are in relation to yourself.
Most approaches in will probably be to the ILS/DME (if you're watching approaches at Tynemouth it will be for runway 25), but I suspect most of these approaches will be under radar direction from Newcastle Air Trafic Control and so will follow the most convenient routing as has been said above, rather than the directions published on the plates as these are for a slightly more complicated and time-consuming procedural approach without radar vectoring. It's this I think you're most likely to see.
Alternatively, in very good visibility, ATC may offer the larger commercial aircraft the option of a visual approach, which although it will be roughly similar to the instrument approaches, to a viewer on the ground it may appear that the aircraft are taking a different route to normal. Most smaller non-commercial aircraft tend to fly visual approaches anyway.
Maybe you'll see me chugging overhead one day too.
(Edited 'cos I'm a techno numpty!) 