I reckon you might have the old half inch pipe, not the 'new 15mm - they are not quite the same.
I'd cut the pipe with a pipe cutter (as above^^), then fit a compression fitting T, and then you can connect up the washing machine as you wish.
You'll need a modern 15mm compression T piece, but if your pipe really is 1/2", you can get 1/2" olives (the sealing ring in the T piece). They are a bit thicker than the 15mm olives - which will come with the T piece. The 15mm olives should work though (especially with a bit of plumbers putty in the joint).
So, buy a 15mm T piece, and some 1/2" olives, and then whatever pipe you really have you'll have all the options covered.
The outlet to the washing machine can be a bit of new 15mm, with the washing machine tap on the end. Simples.