You need a 15mm equal tee, just like the white ones you can already see coming off from the vertical pipes. They look like JG speedfit ones IMHO – get them from Screwfix.
You’ll need some copper pipe OR plastic pipe (but you need to put inserts in the ends before connecting plastic pipe) coming from those isolation valves (which incidentally appear to be missing their locknuts and olives). On the cold pipe put an equal tee on so you have a feed for the washer. You’ll need a washer valve ideally which will go on the end of the pipe coming from the tee so you can screw your washer hose onto.
The pipes going to the sink – you will need to buy 2 bib taps (outdoor style taps) and 2 90 degree wall plate elbows to screw them into. You then take your pipe up to them and install normally with a compression fitting. Don’t forget to PTFE the bib tap threads so it seals into the wall plate.
Taps: http://www.screwfix.com/p/outside-tap-with-hose-union-15mm-x-mm/41948?_requestid=322153
Wall plate: http://www.screwfix.com/p/p803wpcp-2-wall-plate-elbow-x/87929?_requestid=322884
Equal tee: http://www.screwfix.com/p/jg-speedfit-pem0215wp-equal-tee-white-15-x-15-x-15mm/50422?_requestid=323359
Pipe: http://www.screwfix.com/p/wednesbury-copper-pipe-15mm-x-3m/98683?_requestid=323767
Washing machine valve: http://www.screwfix.com/p/washing-machine-valve-without-check-valve-15mm-x/51231?_requestid=325406
Elbows – might need these to route pipe around corners, would need for copper as well as plastic pipe really: http://www.screwfix.com/p/jg-speedfit-pem0315w-elbow-white-15mm-10-pack/57953?_requestid=327110
Pipe clips to fix pipe to wall: http://www.screwfix.com/p/talon-15mm-talon-hinge-clip-white-20-pack/65456?_requestid=326384