Solved my mystery of water pipe freezing!
I added 2nd floor to house way back in 1987 including a full bathroom - plumbing had to be installed about a foot inside the outer wall of house.
The hole in the basement ceiling where the 2nd floor water pipes go up had wads of insulation stuffed in it. That prevented the heat from the basement from entering that area - allowing the pipes to freeze. I removed the insulation using a hooked coat hanger - had just enough room to blow a hair drier into the hole for 10 minutes - water started flowing again. DO NOT put back the insulation! Anyone with a similar problem should check this area and make sure that heat from basement can reach the water pipe column as it goes up to 2nd floor.

Yeah but now you have a unblocked chimney going from basement to 2nd floor, a serious fire risk.

Perhaps a different material of insulation, such as foam. As suggest a fire-stop, is a must.
Best Wishes

ANd paper in the opening was a good "fire stop

the hole for 2nd floor plumbing was nowhere near any chimney.

Chimney meaning uninterrupted vertical column/opening going from basement to 2nd floor. Fire in basement will be able to spread directly to 1st and then 2nd floor w/out your knowing. That's a no-no in construction without something to block it such as fire-block caulking.

If your pipes were run 1 foot inside of the exterior wall as said in your original post and still froze, then there are other air leaks causing the freeze, not just the spot in your basement ceiling.

Points well taken re: fire hazard - this is a 1951 house and likely has other problems.
(I'm not a construction expert or plumber)
Let me rephrase my experience - during last week's below zero weather, the upper floor bath had no water flowing - hot or cold:
Background: I added a 2nd floor to my home 25 year ago, adding a full bath. The only place the contractor could run the plumbing was through an existing plumbing wall, continuing upward to where new 2nd floor bath was located. He had to cut a hole to fit hot & cold water lines, plus the toilet stack, about 5x7 inches. The location was about a foot inside the existing outer wall of house - prone to get colder air, but couldn't be be anywhere else.
The open joist basement ceiling was what I was referring to when using hair dryer to blow heat upwards along these pipes for roughly 10 minutes, to thaw the upper water pipes. It worked. The opening was just enough to allow the 1" hair drier outlet to fit. It had some fiberglass insulation in it. I felt the cold air within this area - realizing that the basement air couldn't enter this opening. So I did not replace this small wad of insulation, in order to allow this air to enter. It got even colder the next night and there was NO freezing of these pipes at all.