Advice on building cavity walls

Speed wise, block and block will be quicker. If you can calculate the area you need to build you can easily cost it by talking to local builders merchants. I have never been a fan of the render idea, I have seen it fail too many times and can be a real menace to paint unless you get a treated one that does not need painting. You will need to chat with the local building regs folks to sort out the size of the cavity and insulation values too. No point making the cavity the wrong size for the sake of a phone call. (Assuming you have not already got information from the planners and building regs folks.) You are likely to need permission for an extension.

Block walls are by far quicker and cheaper than facing with brick.a render finish(if done properly) will last for many,many years with very little maintenance other than painting and also provides a very good barrier against water ingress,in my opinion brick looks better but block and render are far cheaper