How to treat brick efflorescence

Three inch gaps seem quite large - even for a 300 year old wall.

You just need to use a good mortar mix (sand and cement) on the narrower gaps.

Clean out all the old, lose material first to make sure that you get good adhesion and then push the mortar right into the gaps as far as it will go. Work on a small area (1 m²) at a time.

Adding pea gravel to give the mix more "body" might work on the wider ones.

However, I'd get a professional to have a look at the wall first to make sure you're not going to have to redo this work in a couple of years time. If the foundations have gone it might need something more substantial work doing.

Is the wall part of a house or just a garden wall? If it's the latter then dismantling and rebuilding might be the best long term option.

You might want to try a water repellent like MasonrySaver® to seal the brick. Since you are in a climate with freeze thaw cycles, you'll want to be careful which type of sealer you use. If the sealer locks water in the brick you could end up with damage to the brick, MasonrySaver says it allows water vapor to escape so this should not be a problem.

You might want to check with some local masons or home builders, to get recommendations on products that are suitable for your area. If this is a newer home you could contact the builder and ask if the brick was sealed, and with which product.

*I am in no way affiliated with MasonrySaver®, it is just one example of a brick water repellent that I found with a quick Google search. I'm sure there are other (possibly superior) products available. Do your research before using any product.