You can start with something like this:
(setq-default org-display-custom-times t)
(setq org-time-stamp-custom-formats '("<%a %b %e %Y>" . "<%a %b %e %Y %H:%M>"))
This will give you
<Thu Jan 26 2016> for date timestamps or
<Thu Jan 26 2016 11:30> for timestamps with times. If you want zero-padded dates instead of blank-padded dates use
%d instead of
%e above (e.g., if you want
09 instead of
9, but I figured blank-padded would be better if you figure out a way to add the ordinal suffix). For a comprehensive list of the formatting symbols, see here. You can try to combine this with the solution here as suggested by @KaushalModi, but I haven't gotten that to work yet.
Also, if you use custom timestamps in org-mode, be aware that there are sometimes undesired side-effects: see this question and answer and this page from the org manual.