Back in the 1970s there would be strikes at NYRA where the mutual clerks were out and they would race at Belmont anyway. There was a NYC OTB an exit or 2 north on the Cross Island and if you hurried you could get your bets in between races & make it back to Belmont.
The closest Nassau OTB was on Hempstead Tpk in Franklin Square, I used to go to the one in Hempstead many times when I lived in South Hempstead.
It looks like the one in Franklin Square is still there. [www.google.com]
The ones in Hempstead & Baldwin (which I could walk to from my parents house) are gone