1:45 in SIN is plenty of time - you likely will have more than 1 hour to spend in the lounge.

If you are on seperate tickets and miss a connection it depends of the type of ticket. It is the same as if you are coming too late to the airport (traffic jam etc).
If the ticket is flexible you will be rebooked.
If not it depends. In my experience most non US airlines will put you on a later flight in this situation - even if the rules say that they don't need to. Maybee somebody has experience with this on BA.