Let us say I am flying at FL100 with heading 360 and I noticed that ATC just cleared another aircraft to climb to FL100 with heading 270, and we may have a converging path if we proceed with our current flight paths; so, if, from then on, neither aircraft has gotten any further instructions from the controller, at what point should I remind ATC that I may have conflicting traffic based on my current flight profile and the clearance given to the other aircraft? For instance, would I have to wait until ADS-B alerted me of a loss of separation between me and the other traffic (say, coming within 3 nm of each other)? Or I could report the possible conflict earlier?
Generally speaking, you should trust ATC. As a pilot, you do not have the full picture. Just because two aircraft are cleared to the same level, does not mean they are going to get anywhere near each other.
As for using TCAS or similar, this article describes an interesting incident, where one pilot decided to make a turn, based on his TCAS display, which resulted in a loss of separation. If the pilot had followed ATC instructions and not turned, there would have been 15 NM between the two aircraft. Instead, the separation was reduced to 2 NM! The problem is that a TCAS display only shows the relative position of other traffic, which makes it hard to determine the direction and speed of that traffic. For that, only a stationary radar station, like the one ATC uses, will work correctly. TCAS is NOT radar!
would I have to wait until ADS-B alerted me of a loss of separation between me and the other traffic (say, coming within 3 nm of each other)?
ADS-B cannot warn you about a "loss of separation", only a risk of collision. Your ADS-B has no idea what the separation minima are at your current location. If flying across the ocean with no radar coverage, the minimum separation could be 60NM. If flying close to an airport, within visual range of the tower, 1-2NM could (in some cases) be legal.
Please understand that ADS-B, TCAS or similar technologies are not meant to replace ATC. They only work as a secondary safety net, designed to kick in when ATC fails.
With that out of the way, if you find yourself in a situation where you are completely sure ATC has messed up and you are on a collision course with another aircraft, of course you should react. Ask ATC to verify your clearance. If you see the other aircraft, follow the rules of the air. But, to be blunt, I would have a very hard time imagining a scenario where you would notice a conflict before ATC and come up with a better solution than ATC. Don't be the guy who questions everything that is said on the frequency. If you felt like you were in an unsafe situation, call the tower (on the phone) after landing and hear their side of things before you jump to conclusions.
I think you will find there isn't a correct answer for this, and that answers will primarily be opinion based. That said, my opinion is that if you have ADS-B, be patient and let ATC do their job.
Speak up eventually, of course. Especially if you see a clear converging path on your scope. The 3nm you suggested seems like as good a point as any to mention it.
It's a little different if you are VMC. In that case I might wait even longer and attempt to acquire a visual first. That way your reminder to them can consist of your assessment of the need to deviate, or declare that you can maintain visual separation.