Eligibility requirements for the US Grammys 2016 were…
What are the eligibility requirements?
For the 58th Annual GRAMMY Awards, albums must be released between Oct. 1, 2014 and Sept. 30, 2015. Recordings must be commercially released in general distribution in the United States, i.e. sales by label to a branch or recognized independent distributor, via the Internet, or mail order/retail sales for a nationally marketed product. Recordings must be available for sale from any date within the eligibility period through at least the date of the current year’s voting deadline (final ballot).
These days, it can be quite difficult to tie down the exact date of release of a song as a Single, & as Thinking Out Loud was already on an album & had been first promoted live on TV in the UK in May 2014, then as a download from June, the original release to radio was in August [Australia] with official release there in September.
The video was released 14th October - which might mean that someone used that date as an excuse to squeeze it into this year's nominations, but I can't find any evidence of its 'true' US release date.
The first vinyl release was 28 Nov 2014 as Part of the Black Friday Record Store Day
Grammy org : Voting Process FAQs
Wikipedia : Thinking Out Loud
As pure opinion, I think the Academy may have slightly bent the rules in respect of the song's extraordinary performance over the past year…
In June 2015, "Thinking Out Loud" became the first single to spend a full year in the UK top 40. In September 2015, it also became the seventh single to have achieved triple platinum in the UK during the 21st century. In October 2015, the song became the first to be streamed over 500 million times on Spotify, and is also one of the most streamed songs in the UK.