It’s probably best to avoid that section entirely. From Union St., you could turn R on Franklin and take the various side streets over to S. Washington St. If you turn L from Franklin and R on Green St. and L on S. Washington, then it’s just a couple blocks south to the Wilson Bridge. If you’re not comfortable with riding on Washington St., then you could ride on the sidewalk for two blocks.
That’s the route I took a couple weeks ago. That was actually the “official” detour that day. The signs indicated that the MVT was closed just beyond the connection to Union St.
Even if the section between Union St. and S. Washington St. is now open, I’ll probably take the Franklin St. detour until the trail has been repaved under the bridge.