From their website it seems that they use a 3D cavity approach to quantum computing, which is (or at least was) pursued by IBM as well. Here, instead of using superconducting resonators on a chip, you use superconducting 3-dimensional cavity resonators (typically made from Aluminium) in which you place "naked" qubits that are fabricated on a Sapphire substrate. The advantage of this is that the attainable quality factor of the cavity is much higher than for a coplanar waveguide resonator (although those resonators have caught up quite a bit thanks to work that was done e.g. by the Martinis lab), and that there are fewer noise sources around the qubits.
Personally my current bet would be on Martinis or Schoelkopf to win the race, as they have the best access to both funding (though I don't know the funding details of QuantumCircuits) and Academic resources. Most of the original work on Transmon-based qubits was done at the Schoelkopf lab, and it seems that he has excellent technicians on his team.
In general though it still seems more probable that a ion-based quantum computer will be the first "real" quantum computer, as the technology is more mature and (for small numbers of qubits at least) easier to scale.
Exciting times!