BA permits a F/O to call stop during the take off roll for any one of several clearly defined failures or malfunctions, some of which may depend on the aircraft speed.
If the F/O is also the handling pilot, they can call stop for a few other reasons as well. All other failures are decided upon by the Captain, who will make the stop/go decision.
We also let our F/Os hold the thrust levers on take off, taxy the aircraft, and make decisions about fuel, weather, alternates etc.
In our experience, training our F/Os to exactly the same standards as Captains, requiring them to maintain those same standards, whilst treating them as professional aviators and not just wheels and wireless specialists, seems to produce the desired result.