As with all things....it depends...
The colder the air, the more power the engines make (simple mass transfer, and the denser the air, the more mass is transferred per second). Increasing speed increases thrust in some airplanes (F-14 and F-15 among them) again, due to mass transfer. Higher speed = more air rammed into the engines by intakes designed to do just that.
Thrust/weight ratio depends both on thrust, and, of course, weight. Load the airplane lightly, and your ration improves.
The Streak Eagle - lightened as much as it could be, to the point of foregoing paint, with no weapons or extraneous external racks, and with only the fuel needed to set the climb record, took off from Saint Louis on a cold, cold day and set some incredible records for time to climb.
At roughly 450 KIAS, the F-14's intakes boosted engine thrust by about 30% over static levels. So, take an F-14B, with 23,800# per engine, and run it up to 450 KIAS, and you've got 32,000# of thrust, per engine. #64,000 of thrust in an airplane that weighs #42,000 empty. Depending on fuel load, it will accelerate while purely vertical. It's an awesome thing - to see 90 degrees of pitch, and see the airspeed increasing at the same time.
But the air gets thinner with altitude, and so the airplane loses thrust as it climbs. At about 25,000 feet, in the F14B, the airspeed begins to decrease, slowly at first, but then more rapidly as the air gets thinner.
Google the YouTube video for the Streak Eagle. It's an unbelievable performer.