England don't have a team nickname. At the RLWC 2013 opener, the nicknames of all the clubs were on inflatable rugby balls inside the Millennium Stadium... except one and that was England.
In fact, no major English national teams have official nicknames that I can think of. "Three Lions" for football, "Red Rose" for Union. We do tend to have nicknames for our A teams though ("Knights" for League, "Saxons" for Union, "Lions" for cricket).
In fact, we have nicknames for our teams lower than that, with the Lionhearts at community level.
England RL have never been referred to as the "Lions" except by people who cannot distinguish the difference between England and GB.
As for GB being the best recognisable brand... We've had bigger crowds as England in this century than GB, but to mention the growth of the other home nations. England all the way.