I'm going to add to this conversation, not as a wise genealogist but as someone who has spent a fair bit of time in Singapore.
I would be inclined to think that your Gt granddad's birth in Bradford may be the thing he lied about, rather than a birth overseas. If he was keen to get into the army, he may have omitted the Singapore birth because it could raise a whole lot of questions he'd rather not waste time answering.
If his dad was a collier then the dad may have been a stoker on a ship at some stage. Alternatively the dad might have served in the army too, and been stationed in SG for a while.
SG was absolutely booming economically in the 1880s so a strong British presence there was desirable. That could mean military, shipping etc.
Modern Singapore has been collecting birth, marriage and death data for some time now. It also has a growing genealogist community. I'd like to suggest you send an email off to Singapore's National Library for advice on who specifically to correspond with. Either that or use Google to track down the registry of births or whatever they call it there. You may be pleasantly surprised by what you find.
-David C
