Folks,
I have had a few tyre failures over the years, with one exception, there are two common matters, we didn't know we had had a tyre failure, and we didn't have any indication of some time quite serious airframe damage until after landing.
Aircraft were variously, B707-300, B707-320, B767, B747 Classic and -400.
What you do is a command decision, if I thought I had serious U/C damage I would not be retracting the gear, unless terrain (or something else) was a greater danger.
The exception was a tyre/wheel failure on pushback at Dubai, we knew we had a problem when the remains of a tyre rolled past the cockpit.
Tootle pip!!