Usually only some softboxes let you swap speed rings. If the ring is held to the umbrella frame by visible screws, those screws secure the speed ring while still allowing it to rotate, and on some designs the ring can be exchanged. But based on the construction you described, your older softbox likely uses a different design and probably isn’t meant to accept a different speed ring.

So in practice, it’s unlikely that your current softbox can be cleanly converted to Bowens mount.

The simplest workaround is to use an S-type bracket that accepts Bowens-mount modifiers. That lets you use Bowens-compatible softboxes or umbrellas with a speedlight, without modifying the original softbox itself.