The question is not how Gary knew the Adele song Hello"...
Firstly unless anyone wasn't paying their full attention, he knew it because he first heard it in the taxi when his daughter offered him a lift. He asked the cab driver to turn the song up on the radio. I'm no Adele fan myself, but obviously he had never heard of her. She released her first album in 2008. This song was released in 2015. Gary was missing between 1999 and 2016.
As we already know he first heard it in the taxi, the real question is how did he learn it so quickly?
I guess the BBC couldn't fit in all the answers in a 30 minute episode not knowing whether it would be popular enough for recommissioning it, so it was left to our imaginations.
It could be that he just has a photographic memory for lyrics and a fantastic musical ear. But I am geek enough to want more sense from such a great show instead of suspending belief, and that photographic memory theory just doesn't work for me.
Well first of, Gary told Yvonne and Ron that he couldn't get a bus back as he only had old money. So on that notion, he had no modern currency to buy the song manuscript from a music store.
But then Gary is very shrewd. After all, he used to have an antique shop back in the 1990's and so knows a thing or two about collectables, even three now. So it makes sense that he would resort to selling anything he brought with him from 1962, as after all this was a continuing theme in the original series.
But what could he sell? He was still wearing the same clothes when he travelled back to 1962 (who'd want his silly hat anyway?). I didn't see anything else on him which he could sell.
My theory: He sold some of his old money to collectors.
OK, old coins pre-1947 are worth a little more because they have real silver in them, but it's highly doubtful that he would still have any on him in 1962. Unfortunately coins post-1947 are only worth their face value, so unless he had a complete set of coins from a particular year (which is pretty doubtful) then he isn't really going to get much at all for them, and would likely take a lot longer to sell anyway, unless he had very large pockets.
This leave banknotes. Fortunately this is where I think we can stop suspending belief, as old British banknotes are worth a lot more to collectors. Just check eBay.
Also it is plausible that Gary may still have coin/banknote collecting contacts in 2016 who used to visit his store up until 1999 (assuming that he naturally had some). He'd only need to sell maybe a couple of old 1960's banknotes or maybe even just one to afford the song manuscript.
But even if this is the most sensible explanation. It still leaves me with another question which I'm unable to fully answer.
What was so sensible of Gary putting such trivialities like learning this song first, above finding out if he can even return back to 1962?
For all he knew the time portal may close at anytime again. But this is Gary Sparrow, I guess he never really did fully understand the word "priorities", doh.