How close would you dare to get bias slick to a wheel opening?

1” is a safe bet. That’s as close as I want to be.

Middle finger width while using caution on turns
More clearances are better, especially for street driving

On my Valiant I can barely squeeze my fingers between the inner fender and the sidewall. They have never touched. Coilover/ladder bar car.

On my Dart I had a knuckle's worth, a tight one at that, of clearance between the leaf and the rolled inner fender lip. I was running some tall sidewall bias plies and the sidewall would just rub the side of the leaf pack in hard turns. If I had a shorter/stiffer sidewall it would not have hit at all.

Slicks tend to get taller and narrower as they are spun up so in a straight line you'd get a bit more clearance. In turns, well go slow.

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Are you asking about clearance to the wheel opening ?

Or the side walls to the inner or outer wheel house and/or leaf spring ?

It depends where is the tight clearance. Inner wheelwell needs miles of room due to articulation. As the axle articulates, the tire travels in an arc towards the center of the car.
To the leaf can be very close because they move together.
Tire to outside at top can again be close because not much is going to push it towards the body.
The leading or trailing edge of fender to a slick needs room because they grow with rpm.