Still reading it, so the answer might be covered late in the campaign but I haven't seen any mention of what actually HAPPENS when the Temporal Trace Locator actually finds the traces. Do they transform into little objects like the pieces of the Key To Time? Does it suck the trace into the locator like the traps in Ghostbusters? Does it somehow patch the scar? Or does it just go DING cause there's stuff? Do they have to take the source of the trace with them? (The second one's a part of the outer hull of a space station, how do they get that off? What do they replace it with? Do they have to contend with having a piece of Axionite growing tendrils and trying to take over the TARDIS every episode?)
"We can go anywhere and any time in the universe. But it will probably be London during the Blitz." -Inspector Spacetime
The Locator collects the traces from each of the objects, not the objects themselves. As the traces are hyper-dimensional scars in the Vortex, they aren't physical objects - presumably some sort of energy residue instead. After each adventure, the Locator is plugged into the TARDIS console to download the latest trace in order for the TARDIS to analyse it and begin to correlate the data with that of the other traces the players have collected.
No problem. I'm not sure that it's clearly stated anywhere, but between the description of the Locator on page 38 and the Branch Point text on page 160 (which you probably haven't got to yet) it seems that's the intent.