Blue Spot from Amazon UK are fine.
The core itself should be cylindrical with the threaded boss in the centre - if not it has been damaged (oval) most likely by dropping. The arbor likewise should be concentric with the drill shaft & core. Otherwise the core will turn in a precession making cutting much slower and getting started even with a guide drill very difficult indeed :-)
If your drill is a chuck (vs SDS), remove it from the drill and turn it 120-degrees, check again, then repeat if necessary. Sometimes the chuck does not take to the arbor "hex" and if pushed too far into the drill the chuck will not self-centre.
Also verify the guide-drill which is often a "spike fit" is in fact central, they can become canted in the arbor requiring a few whacks on the drift to get it out and relocated. You must use a guide drill if the drill is of any size.
A new sintered diamond core can be glazed, which makes starting the first time doubly difficult. This can be solved by turning it manually with the guide-drill in place or running on an old paving slab slowly. The variation in cheap diamond core drills is usually the amount of diamond and its even distribution (or lack of) through the sintered tabs. That said, cheap bits work fine - it is only the really tough materials that need "proper bits" otherwise you will see nothing but sparks and take many hours.