Buy a plunger , hold a cloth over the overflow vent and then use the plunger with a vengance , you will get a load of gunge out of it , and it will stink . Then use bleach to clean the pipe .
If the plunging fails then a product called ONE SHOT is probably the best on the market. It is used by the professionals so be warned and wear hand and eye protection, as it is a high acid content and eats through just about anything yet does'nt hurt you're waste pipes, available at most plumbers yards. You can get it from B&Q aswell but it is not the industrial strength from there.
As an alternative to the traditional plunger, have a look at Screwfix.com website, do a search for product code 13894.
This is an inexpensive high-pressure water pump.