This is almost like asking "Chevy, Ford, Ram, or Toyota?"
All of the above (.270 wsm, .300 wm, and 7mm) are all great choices. You can also look at the .300 WSM, .270, .260, .280, .280 AI, .308, .30-06, 6.5 Creedmore (if you want to go "exotic"), 7mm-08, .300 RUM, 7mm RUM, 7mm WSM, etc....
I've seen both elk and deer taken with .243 and .25-06, but I have a hard time recommending them for elk, personally. I've seen big bulls dropped to all of the above-mentioned cartridges, plus many, many others and at ranges from 30 yards to 600+ yards. A friend of mine's son killed a giant 5x5 bull with his 7mm-08 at 400 yards and a solid mule deer at closer to 500 yards. The bottom line is find a good rifle with good glass that you can shoot well and not get recoil-shy with. If you already have good choices at home, start there. If you have to go buy new, look at the above choices and you can pretty much pick one from a hat and win.
If I had to pick one of all the above mentioned, I'd look at either the .300 WM or .300 WSM to save some weight, but that's probably because I own a .300 WM and love it. If I was starting new, I'd go with the .300 WSM to get the exact same performance in a lighter weight rifle. The only concern with the .300 WSM is if you hand-load. The magazine length in most rifles will limit what you can do with bullet seating depth adjustments to fine-tune the accuracy. You'll be able to seat your bullet way farther out than the magazine length will allow. I'd probably even go with a .280 AI just to go against the grain a little and continue my streak of being different.