I have a UK short term standard visitor visa issued in USA (I am an Indian national) and I'm traveling to Dublin Ireland via London (and I don't have a Ireland visa). Will I be allowed to board the flight with my UK Visa and will I be allowed to take my connecting flight to Dublin (short stay visa waiver programme)? Any help would be appreciated
You are not only allowed to fly this route, it is mandatory that you do so.
In order to use the Irish Short Stay Visa Waiver, you must first clear immigration in the UK, then travel to Ireland. You cannot take a flight directly to Ireland without going through the UK first.
This doesn't apply to exiting; you can exit via whichever country you wish.