My Wemos boards have been running for, in some cases, many weeks now, waking from deep sleep every 15mins without problems. I connect D0 to RST with a ~500R. I did experiment with using using 4K7. This did not work and resulted in what looks like the same situation you are facing where the deep sleep ends, a line of garbage appears on serial monitor, but the sketch does not run. I then tried 2K2, which worked well on 5V USB power, but did not work on ~3.8V battery power (3xAAA NiMH). Finally i used 510R which runs fine at either voltage.
I have also experimented with light sleep and modem sleep. I have not been able to achieve low current consumption with light sleep. In fact i found it used more current than modem sleep. In my circuit i need to monitor sensors every 2~5ms, so cannot light sleep for longer periods, which might explain my findings.