I recently dislocated my elbow. How long will it be before I can do pushups again?

It depends. This depends on the severity of the injury, the number of disrupted ligaments and associated injuries and should be discussed with the orthopedic surgeon treating your injury. Generally, stability of the joint will progress as ligament healing occurs over 6-12 weeks. If instability persists, surgery may be necessary, and this can prolong healing time and return to activities such as push-ups.
Dont rush it. Heal what is the hurry? Work on rom first then strength. Full extension is often lost foreverwait six weeks then work with ot to get motion back then strengthen the elbow is uniquie muscle strength will not help with stability it is all ligaments and boney anatomy luckily it does not have a complex motion except forearm rotation.