Peel cucumber and cut in half lengthwise; scoop out and discard seeds. Chop cucumber; place in a large bowl and add tomatoes, bell pepper, olives, lime juice, broth, garlic, onions, and thyme. Stir well; season to taste with hot pepper seasoning. Cover and refrigerate until cold (at least 1 hour) or for up to 1 day.
To serve, stir gazpacho well. Ladle into bowls and serve with lime wedges.