Definitely a waterproof jacket for everyone; that can serve as a windbreaker, which is also likely to be useful. I was glad to have waterproof shoes, but it probably won't be so cold that your feet will absolutely freeze if they get soaked, and I guess you probably don't already have waterproof shoes for the family. That's not an expense I'd want to incur if I lived in a hot, dry climate and was making just a brief trip to Northern Europe.
I also used a merino-wool underlayer under my slacks because I am cold-natured and I knew I'd get wet below the waist since I had only a rain jacket with me. Worn under 97% nylon slacks, that extra layer helped keep me comfortable on chilly, wet days. The nylon slacks tended to shed water rather than absorbing it, but they still ended up somewhat damp. (As for denim, what would soaking-wet jeans feel like when the temperature's 52F/11C?) Note that I was in Norway for nearly 3 weeks, with many more weeks in Sweden and Finland, so I had a lot of potential exposure to cool, wet weather to consider.
Norway--especially the west coast (including Bergen)--gets a great deal of rain. On rainy days the temperature can get rather cool--and Bergen is rarely really warm because it is overcast so much of the time, so everyone should have a warm layer (like a light fleece jacket) that can be worn under the rain jacket.
To get a good idea of the range of weather I may experience at a particular destination, I like to check the actual, historical, day-by-day weather data on the website timeanddate.com.
Bergen weather July 2022
Oslo weather July 2022
I've linked you to the data for July 2022, but it's really important to check multiple years, because weather is highly variable. I normally look at the most recent five years' worth of data. You'll want to check early August as well. Use the pull-down box at the right above the graph to change the month and year displayed.