The same bolt works with the Lapua case you ust need a differant bolt head.
As for their being more meat around the chamber , the standard Savage has a barrel OD of 1.062" (or realy close) and the large shank has an OD of 1.125" so yes their is roughly 1/16" more meat around it , yeas both have nuts and the nuts end up having the same OD just in front of the action. I personaly feel that the material added by the barrel nut doesen't ad any strength to the chamber area as the threads are never a very good tight 3A fit so their will be some room for the barrel to expand a bit before the barrel nut starts to take up any of the pressure , sme goes for the action threads. When the barrel is fitted like a Remington's this allows a full 1.25" of meat around the chamber. When an action is trued the threads are recut to allow perfect alignment with the bolt race then the threads are cut on the barrel shank just slightly over sized so that the action can be test fitted as the final threads are finished this allows for the best possible thread fit and maximum pressure.
The remington 700 is threaded 1.062"x16 TPI , the Savage is threaded 1.062"x20 TPI making it have shallower threads than the Rem so technicaly the Savage reciever desgine is stronger than the Remingtons as it allows more meat in the action walls. The Savage bolt lugs are generaly a little thicker than the Remingtons also adding yet more strength plusss the Savage seat the shell case a little farther into the barrel due to not having a bolt nose countersink like the Remington.
All that said I know a guy that had a close call with a Lawton action shooting a wildcat on the Lapua case , the Lawton is the same spec as the Remington and this action was SS which is not as strong as a chorme moly (blued) action ,the case (7mm-338Lapua) was mistakenly loaded with a powder thats ALOT faster than whats normaly used , the end result was a 180gr Berger run out at neer 3600fps , a bolt locked down and a bulged barrel ahead of the recoil lug , the barrel was removed and the action showed no sign of damage and lives to fight another day , the barrel was ruined. The load was run through the "Quick Load" program and was said to have a pressure over 150,000 PSI , that double what the factory specs are and I think the only whay that this can be done is with a big mistake in the powder choice.
In short I think that the Savage is perfectly safe for the Lapua case if its put together properly , but that goes for any action.