The book The Artillerist's Manual By John Gibbon, from 1860 gives us some insight into what was in use at the time. A clip from the appendix page 13 shows the components of an ammo box distributed with 12-pound guns and Howitzers.
The last three items on the list are included to provide various means of firing the weapon; the friction primer, the slow match, and the portfire.
The friction primer was probably the primary item used by this time to detonate the cannon charge. Note the packs of 48 and 60 included in these packages. These were self-contained and packed in their own container to keep dry and ready. They operated by friction, so a pull on a lanyard would set off the priming powder. No external spark necessary. From a NPS website:
The device was a copper tube (fig. 19) filled with powder. The tube
went into the vent of the cannon and buried its tip in the powder
charge. Near the top of this tube was soldered a "spur"—a short tube
containing a friction composition (antimony sulphide and potassium
chlorate). Lying in the composition was the roughened end of a wire
"slider." The other end of the slider was twisted into a loop for
hooking to the gunner's lanyard. It was like striking a match: a smart
pull on the lanyard, and the rough slider ignited the composition.
Then the powder in the long tube began to burn and fired the charge in
the cannon. Needless to say, it happened faster than we can tell it!
The next two items would have served as backup in case of problems with the friction primers or they ran out.
The slow match was an older technology, and might have been more susceptible to moisture issues, as it required direct contact to some form of cartridge fuse or priming powder. From the same NPS website:
Before 1800, the slow match was in universal use for setting off the
charge. The match was usually a 3-strand cotton rope, soaked in a
solution of saltpeter and otherwise chemically treated with lead
acetate and lye to burn very slowly—about 4 or 5 inches an hour. It
was attached to a linstock (fig. 18), a forked stick long enough to
keep the cannoneer out of the way of the recoil.
The portfire, the final entry, would have been a failsafe for all-weather use, as it was used in naval artillery where constant water exposure would have been an issue.
An glossary entry from the website civilwarartillery
PORTFIRE: A composition of nitre, sulphur, and mealed powder driven
into a case of strong paper. Portfire was used to fire guns previous
to the introduction of the friction primer. It was also used in
emergency situations and to fire rockets. A full length portfire would
burn for about ten minutes and could not be extinguished by water.
So of the several options discussed in the 1860 book linked above, most had some degree of functionality in adverse weather conditions.