Chuck;
Cracking and hardness in the heat affected zone as discussed above can be caused by several issues, and a most common issue is lack of, or not enough pre-weld pre-heating. And of course if one does not know the exact content of the base metal selecting filler metal is a stab and if the selection is wrong cracking can occur. I have not seen any such problems with tig welding our typical English made barrels of the pre-WWII era, however they are very soft. I do not recall what "E" series steel was used on barrels of that time, but I will ask one of my colleagues who knows and get back to you. However, barrels of that time frame in England were not of chrom-moly, and I suspect that the barrels you reference must have been American made high strength barrels.
As to receivers cracking from welding(I assume that you are speaking of double gun actions that have been color hardened), it would not surprise me that they would crack if they had not been annealed prior to welding.
I spoke with one of my colleagues today about the issue raised above(he is a professional welding engineer with nearly 50 years experience) and he said that although he has never been involved in welding shotgun barrels, pre-heating to about 350 degrees F would be his choice on welding a tube that is as thin as a barrel.
I do know from some of the older Birmingham gun tradesmen( the one I am speaking of died in 1957) that they(he) learned to weld with OXY/ACL when it became commonly available to the trade around WWI. He welded actions and barrels much to the dislike of the proof house--but try as they may they could not get his welding to fail. Different era and different metallurgy though.
Bv