Surely therefore our locally measured time is not the cosmological time t but rather the conformal time T?
I don't think so. Our locally measured time is our locally measured time. If you had a clock that started ticking when the big bang occurred, the clock reading would be 13.8 billion years. If it displayed the conformal time, the clock reading would be 46.9 billion years, see the Wiki particle horizon article. Perhaps what you're driving at here relates to time dilation, wherein the expanding universe can be likened to pulling away from a black hole? When you're near the event horizon, your clock ticks slow. As you pull away, the clock rate increases. So whilst your clock reading is 13.8 billion years, this is not a true reflection of elapsed time. The outside observer might say that 46.9 billion years had elapsed. Only for the universe, there is no outside observer. And of course the particle horizon is as far as we can see, and the universe might be bigger than that.
All the physics experiments that we perform locally are assumed to occur in flat spacetime...
There is some confusion between flat as in horizontal, and flat as in not curved. See this depiction of gravitational potential:
CCASA image by AllenMcC, see Wikipedia Commons.
The force of gravity at some location relates to the slope of the plot or the spacetime "tilt", whilst the tidal force relates the curvature of the plot or the spacetime curvature. Either way, spacetime isn't flat in the room you're in. If it was, there would be no gravitational field, and your pencil wouldn't fall down. NB: spacetime curvature is "the defining feature" of a gravitational field because without it your plot can't get off the flat and horizontal in the middle.