Probably simplest would be to either solder a "micro Losi" connector to the board, or make a connector lead like Neil explained to convert from JST 1.25 to micro Losi. It's probably the most common connector for those small 1S batteries and you can get the batteries all over the place in a myriad of sizes and shapes. Search on Amazon for "3.7 volt LiPo" and the vast majority of the results will have the (white) micro Losi connectors.
		
		
	 
Thanks Rick M.
I have already ordered some items and I hope I've done right? (If I had seen your post, I would have followed your advice, but too late now, see what you think)
So I have a toy/mini drone that takes batteries (with the required dimensions) with JST 1.25 connectors (Thank you for your identification).
After extensive research I can confirm that batteries with these specs are simply not available, but, same dimension, batteries with JST 2.0 connectors 
are.
How, then to connect battery with 2.0 connector to 1.75 connection on drone.
I therefore ordered:
1) Batteries with 2.0 JST connectors.
2) leads with 1.25 connectors (with male and female connectors)
3) leads with 2.0 connectors (with male and female connectors)
I intend to connect a female 2.0 connector to the blank end of one of the 1.25 connector (male) leads. (After shortening the lead, obviously.)
So this lead will then connect the new battery (with 2.0 JST male connector) to the drone. And the operating battery in the drone will just have a slightly longer lead. To charge the batteries I will obviously remove the made up lead.
Hopefully this will work?