There are similar papers to the one below that state: "Improvement 2 weeks after the injection when the corticosteroid anti-inflammatory effect is engaged predicts long-term benefit." I think there is another out there by Plastaras that has similar statement, but those are mostly for helping us determine longevity of efficiacy. I think that has gotten misconstrued as "benefit will occur by two weeks."
I generally tell people two days for ESI to start working, should continue to improve over the first two weeks (with most benefit in first week), and where your pain is at two weeks is where we expect it to stay for 3 months (hopefully longer) and should further improve with McKenzie PT, lifestyle changes, etc.
Academically my statement may be wrong, but in practice it holds true for me.
Epidural injections of local anesthetic or a corticosteroid are frequently given to diagnose and treat patients with radicular pain originating from any spinal level. The best-quality evidence supports a transforaminal approach in the lumbar spine. ...
www.ncbi.nlm.nih.gov