Bubblz the reason you lag is because you are in the UK, same as me. The server is like 3000 miles away. You may have a good internet connection, but your latency to the US will still be several magnitudes greater than someone living on the other side of the pond.
You will lag more on PVP and survival than any other world. Nether will be the least laggy, followed by build, followed by PVP, followed by survival. If people want an explanation here is why:
Survival: Mobs. Mobs kill worlds with lag.
PVP: Fire spread + lava + water + the actual PVP system itself. The minecraft server has to relay a lot more information to the client when PVP is turned on, and vice verse when the client is sending to the server. This extra information creates increased latency when having something like a PVP fight.
Build: LOTS of animals, massive variety of block types, weather actually interacts with stuff unlike the nether, lots of minecart systems
Nether: It doesn't lag at all. The only thing in the nether that can create lag is the lava, and there is no 'flowing' lava in the nether, only still pools and tubes of lava from the ceiling. There are no mobs, so no lag from there.