‘Turnaround time’ is defined as the date from when a manuscript was first received by the journal to the date the author of the manuscript was provided with a first decision. Obviously turnaround time and acceptance rates vary from journal to journal; this information may be available on the journal's website, or on individual journal articles.
Most journals don't publish their acceptance/rejection rates but some do.
Try searching the journal's web site to see if they publish such data.
New England Journal of Medicine
"We publish only the top 5% of the 5,000 research submissions we receive each year..."
American Psychological Association Journal Statistics and Operations Data
Stats from 2004 to 2016
Some Springer Journals include the following information:
Go to a Springer journal web site.
Scroll down the screen and click on About This Journal under Additional Links
Scroll down to Journal Metrics and click on the Learn More button.
Here you'll find turn around times and usage data.
Turn Around Times By Subject
Here is a small selection of lists of journal titles that provide information by subject area. Some give details of acceptance rates
Backlog of Mathematics Research Journals Stats for 2016
British Philosophical Association Survey Survey data from philosophy journals from 2011 to 2013.
Political Science Journals This data is mostly from 2011.
American Psychological Association Journal Acceptance Rates
Stats from 2004 to 2013
Some high impact/quality journals now require authors to share data used in their articles when readers make a reasonable request.
PLOS journals require authors to make all data underlying the findings described in their manuscript fully available without restriction, with rare exception.
American Psychological Association journals
Ethical Principles of Psychologists and Code of Conduct
Section 8.13 Sharing Research Data for Verification
(a) After research results are published, psychologists do not withhold the data on which their conclusions are based from other competent professionals who seek to verify the substantive claims through reanalysis and who intend to use such data only for that purpose, provided that the confidentiality of the participants can be protected and unless legal rights concerning proprietary data preclude their release. This does not preclude psychologists from requiring that such individuals or groups be responsible for costs associated with the provision of such information.
(b) Psychologists who request data from other psychologists to verify the substantive claims through reanalysis may use shared data only for the declared purpose. Requesting psychologists obtain prior written agreement for all other uses of the data.
The International Committee of Medical Journal Editors announced a new requirement that reports of clinical trial data submitted to ICMJE journals after July 1, 2018, must include a data sharing statement.
Data sharing isn't mandated in journals following the ICMJE recommendations but "editors may take into consideration data sharing statements when making editorial decisions."
List of Journals Following the ICMJE Recommendations
Springer Nature journals
Encourage journals "to adopt a standard research data sharing policy."