Any discussion of lubricants is invariably controversial, but in the
world of clocks many, if not most, have adopted synthetic oils.  Some
of these are theoretically formulated for clocks, though the general
suspicion is that they are expensively-repackaged industrial oils and
greases.

For my clocks I have been using Mobil 1 0W-20, which is a [synthetic]
motor oil you can buy at Wal-Mart or any auto parts store.  It's a
little heavier than the Nye 140 clock oil I'd been using, but it seems
to behave splendidly.

Synthetic oils will not harden, either with age or with low temperatures,
and Mobil 1 is compatible with a large variety of bearing materials.
(Clock bearings are typically brass on steel.)  It's about eight dollars
U.S. per quart, which will last me the rest of my life unless or until
I spill it.

Note that there are several sub-species of Mobil 1 oil -- I think mine
is theoretically for long oil change intervals -- but any of these work
well for clock mechanisms.  I'm afraid I don't know the requirements
for your particular application.

Sewing machine oil is quite thin, and I don't know its long-term
behavior.

Mark Kinsler

 [ A data sheet showing the nominal amounts of phosphorus and zinc
 [ in the various versions of Mobil 1 motor oil is at
 [ https://mobiloil.com/~/media/amer/us/pvl/files/pdfs/mobil-1-oil-product-specs-guide.pdf 
 [ -- Robbie