IMHO, it an unknown..unless it's a scenario where there's such frozen ground that it's obstructing proper drainage. I can't see how outside drainage from roof and downspouts is affecting inside the house.
I'm sure that your plumber friend will have a good fix and answer for you.
Although, when I had my house, I had a private septic system located in my backyard we had a problem that caused a backup. The 'grey water' (water that is not sewage) from the upstairs kitchen disposal, water from the washing machine in basement backed up into the washing machine. it turned out the main drainage pipe just outside of the house basement wall had a break (frozen ground and/or frost heave) just beyond the wall so that the drainage wouldn't get out the last 10 feet to the cistern where the grey water destination should be.
My understanding of drainage would be that your ground might be so frozen that it's obstructing proper drainage. But I can't see how the snow melting itself might cause it.