If I understand what the issue is:
1. The resized Sako brass will not chamber in your gun, but once shot "unsized" Wthby brass will chamber?.
2 The resized Sako brass will not chamber in your gun, but Wthby brass shot in your gun and "resized" will chamber?
Q: Will the unsized Wthby brass chamber in your gun w/o resizing?
Chances are that the Sako brass was shot in a longer chamber than yours; at higher pressure due to the less freebore throat in the custom chamber (high probability). This means the Sako brass will be much longer at the shoulder.
Now one of two things are happening.
1. Either your die is too long to size the Sako brass and is bottoming out on the ram before hitting the shoulders. Not uncommon.
2. You have pushed the brass back too far on the Sako brass at the shoulder/case junction. This will cause a minute bulge that will prevent chambering and is almost impossible to see. This will prevent it from chambering in your gun and chances are the Sako. I did that once on a 7 STW.
I would almost bet this is the situation if your Wthby cases are resized and fitting the chamber.
If you have done that (2), and resized all the Sako brass. You are kind of stuck with it, if it will not chamber.
To test that, get 1-2 pieces of other Sako brass that has not been resized with your die and move the die up at least 3/8 inch off the ram and start back by hand turning the die until you feel the case contact. Stop and resize, test chamber and continue down slowly 1/16th turn at at time until it will just chamber.
IF it never chambers then you are going to have to have someone machine about .015 off the bottom of the die until you can push the shoulder back .002. Occasionally that happens also. An alternative is to buy as set of Redding Competion shellholders which will allow a .010 range at .002 increments with the set of five shellholders.
If your friend reloads, have him resize the brass using his dies and then shoot it in your gun if they would chamber. This would be the quickest and easiest way.
FWITW
Do not adjust the dies on top of the ram and do not set belted cases when reloading to headspace off the belt. BIG no no on both counts.
Belted cases headspace on the belt when fired as factory. Once you reload them you should set the dies up to headspace on the shoulder for long case life unless you just like buying expensive Weatherby brass all the time.
IF you headspace off the belt or on top of the ram automatically, you will end up with excess headspace that way and case separation in as little as 3-5 shots. Those instructions are not meant for accuracy reloading or long brass life as they do not take into consideration minimizing head space.
Absolutely no need to headspace on the belt. If you do, 3-5 firings and you are going to have case separation and you and your gun are SOL until you can get the broken case out of the chamber.
If you try to use your dies to reload for the two guns you will have issues also.
BTW the trick for using one set of dies for two or more guns is to go to sinclair international reloading supplies (or others sell them too) and order a set of Skips die shims. They are round shims in different thicknesses that fit under the die to raise it up without making any permanent adjustments on the setting. That way you set the dies for the shortest brass and use shims under the die when sizing the longer brass.
Skip's die shim kit - (7/8-14) - Sinclair Intl- Cost is $9.95
Now you will have to use the shims to resize the longer brass and that will take some adjustments with varying shims until again the bolt just closes on the sized longer brass.
Be sure to mark down what the total thickness is of the shims in your records for the longer chamber and gun.
BH