Upvote:2
This is addressed in Paragraph 20A of the rules, which says...
Leave to enter or remain in the United Kingdom will usually lapse on the holder going to a country or territory outside the common travel area. However, under article 13 of the Immigration (Leave to Enter and Remain) Order 2000 such leave will not lapse where it was given for a period exceeding six months or where it was conferred by means of an entry clearance (other than a visit visa).
A private medical visa was a type of entry clearance, but it was also a type of visit visa. So your wife's leave will have lapsed unless her second visit is somehow related to her treatment (in which case she will need to show the IO a letter from the clinic).
It's ok to show the IO the entry clearance if he doesn't spot it, but she should present herself as a standard visitor.
Your references to the "180-day visa-free entry scheme" isn't quite right, it's a visa national scheme. Please read Under the VWP a visitor from USA can stay,6 months in any 12,when does the 12 months start?
Comments to your question have pointed out that the private medical treatment visa is now legacy. The current visitor rules (including those for Americans) are at Appendix V.