Actually if I'm not mistaken, I believe the UGLE leaves it up to the GL'S in each state. Not sure though, that's why I'm asking.
That's for recognition. The process goes 1) Local recognition is voted in. 2) MWPHGLofXX applies to UGLE for recognition. 3) UGLE approves recognition. 4) From then on MWPHGLofXX is just one more GL on the UGLE list. It doesn't even matter if the local treaty grants visitation.
For some reason MWPHGLofOK has not requested UGLE recognition even though GLofOK recognizes. So MWPHGLofOK members can't approach UGLE for visitation.
To visit a UGLE lodge the process is the same for any recognized jurisdiction. Get credentials from Gr Sec for foreign travel. Send a letter of introduction your Gr Sec to UGLE Gr Sec to the lodge you want to visit. It may be possible to shorten that sequence if a member of the lodge you want to visit has been to your lodge.
On visitation - If your local treaty does not include visitation, other states may well welcome you for a visit. Some states just plain recognize. Other states have their own treaty. Other states do what your local treaty says. You just have to ask and/or look it up.