""Does the Rome-London/Glasgow-Dublin ticket option seem okay for respected entry in London? "

Sure"

Not true.

UK (and Irish) Immigration WILL refuse entry, as I said earlier, to passengers without an onward reservation to a destination OUTSIDE the UK/Ireland Common Travel Area if they're not convinced the applicant is going to leave. And, BTW, will mark the passport "deported", thus making future direct entry into the "other" country practically impossible.

Why can't she do what your country demands of us: just have a ticket home before she arrives here?

An onward reservation to the Republic is better than no reservation at all. But your offensive nonsense about "Hopefully she may be able to gain entry as would an US citizen.", and her wilful refusal to have a reservation out of the CTA makes it look increasingly likely that she DOES intend abusing our rules. Both of you seem to have this bizarre conviction you can behave here in a way your government would never allow us to in Canada.

Just tell her to buy a bloody ticket home and stop smart-arsing.

Thank you everyone for your knowledge. I was finally able to gain communication last night with daughter and passed along information.

flanneruk...thanks for the 'beware' information.
However, I may have written an earlier message generating a misunderstanding. "Hopefully she may be able to gain entry as would an US citizen" was meaning to be a wishful polite response to xyz123's (Sep 4 at 1:48pm) comment, "Assuming she is carrying a US passport, she is entitled, at the discretion of the immigration officer, to be given leave to enter the UK for six months". There is no "offensive nonsense" intended. I should hope that people asking for information for clarification on this site is not "bizarre" and subsequently mislabelled "smart-arsing".
Wishing you joys in your journeys.