What’s involved in converting a 7mm mag Winchester action to a 375 or 416? I know the Winchester safari actions for a 375/416/458 have been opened up, which is why the two base screws for the back scope mount are so close together.

Would a 7mm mag action have to be opened up in the rear for the longer cartridge? Or is that unnecessary?

I just converted a 7 Rem mag to a 375 Ruger in a tang safety Ruger 77. I chose the Ruger magnyum rather than the H&H so I wouldn't need to lengthen anything, like the bolt stop, magazine, etc. I'm still working on a new stock so haven't shot it yet but it looks like it's going to feed fine.

Other than a new stock, all I would have needed to do was a new barrel and chamber. I realize your action is longer inside than the Ruger 77 but unless you're set on the full length cartridge the Ruger 375 (or the Ruger 416) would be an easier conversion.

It's child's play; remember that all post 64 long actions are made for the 375 length; many do not realize that the pre 64 model 64 was only for 30-06 length and had to be gutted out to fit the 375.
Even a .473 bolt face post 64 Model 70 can easily be converted to 375 length. You remove the block from the mag box. Shorten the bolt stop. Maybe tweak the mag box feed lips. If you start with a 270, then open up the bolt face and extractor. Ideally you need a lathe for that but I've seen fair jobs with a dremel.
Make an angle cut in the bridge; not necessary if you don't want to eject loaded rounds. The receiver itself is plenty long enough so no alteration needed at the magazine end.
A Model 70 is not a Ruger 77. You can't put an H&H in a 77 tang safety. The middle guard screw is in the way.

DPCD covered it all. Like he said, childs play, even I can do it. I do the same thing to make the action work with 7mm mag and 300 win mag and the super long low drag bullets and long throats popular today. Only thing I do different is buy a 375 length box instead of taking out the mag block. You can get them from here for about $35 with shipping:
https://www.midwestgunworks.co...mgwi/prod/U353071450

unless you are just in love with the HH CASE, A 7REM could become a 375 or 416 ruger without much pain