A little further research, I could find no US production rifle ever made in .275 H&H. So, the rifle your friend got was either a custom chambering, or foreign made (likely by H&H).
I could also find nothing on a .416 H&H round. .416 Rigby, yes, but not H&H.
Possible confusion with an H&H rifle chambered for .416 Rigby being referred to as a .416 H&H?
ballistics puts the .275 H&H a bit below the performance of the 7mm Rem mag today. More like the 7mm Express/.280 Remington.
however, I think that is less than the possible potential of the round. Base size is essentially the same as the .300 H&H, but the case is shorter, (2.5" vs. 2.85") but even with the taper will hold 65+/- grains of powder, depending on the bullet used.
The H&H rounds delivered over the top velocities, for the day, but did it without pushing pressure to the limits. They did it with big (magnum) cases, not extreme pressures. The long tapered cases, and moderate pressures had a very practical benefit in hunting, making the cases easier to extract even in African heat.
Original .257H&H brass would likely be Berdan primed, but forming new cases from boxer primed brass should be easily done.