The equivalent of the Fed's fund in th UK is SONIA, not LIBOR. SONIA is for overnight interbank lending, and I believe is "controlled" (with the BoE pushing the rate where it wants by using its unlimited supply of funding to manipulate supply and demand) in the same way as in the US.

But these days pretty much the whole term structure of the yield curve is manipulated by central banks (see Operation Twist for instance).