Ok, well, do you have much experience designing electronics? Just because a PSU is 'rated' for 1A, does not mean it can act as a constant current supply of 1 Amp. If it is voltage regulated then it could shut off from short-circuit protection, reduce its voltage or anything else when too much current is drawn - the point is, you can't just rely on a PSU's inherent limit to make a constant current supply.
Secondly, if you are stepping this up to 16.8V from 12V then clearly the input current would have to be higher for an output of 1 A. P = I*V.
Thirdly you seem to want to just 'turn the damn LED on'... for indicating fully charged. If you don't have much experience with DIY, do yourself a favour and buy a battery charger. Or, at the very least, by some 'modules' like a CC/CV supply board (they go for a few $$ on ebay). These often not only have current and voltage limit adjustments, but also indicator LED's to determine whether its in constant current or constant voltage mode. They also tend to have a low-side current sense resistor which you can use, along with a comparator to indicate when the current draw is below C/10.
Whilst many seem to go over the top with regard to the 'danger' of poor charging practices, these cells really can 'pop'. But for lower charge currents, generally it will just destroy your batteries, which harms your wallet more than yourself.