The short answer
In central Bangkok you're rarely far from a major private hospital with a 24-hour emergency department. That means Bumrungrad, Samitivej or Bangkok Hospital around Sukhumvit, and BNH in Silom. King Chulalongkorn Memorial is the big public hospital in the centre. Traffic is the problem, not distance: in rush hour, a nearby hospital beats the "best" one across town. For a serious emergency, call 1669.
General information, not medical advice. Phone numbers come from each hospital's own website where we could reach it, or from the Australian government's hospital list. Lines marked check before you go we couldn't confirm.