Although I can't find any information on Time Rose performing at The Factory in Birmingham, just an unconfirmed appearance at Birmingham Mothers with no mention of a date, there are two dates referenced for Bonham playing with Chris Farlowe and The Thunderbirds at The Factory on 10 & again on 17 August 1968 along with a further reference to say Bonham would continue to play with them until late August. https://garagehangover.com/chris-farlowe-thunderbirds-gigs-1963-1969/ Unfortunately none of this helps to pin-point a date for Plant's return trip from Pangbourne.

The Top 20 chart dates from the week 11-17 September 1968 according to the information at https://www.officialcharts.com/charts/singles-chart/19680911/7501/ Also, in an interview Page gave to Chris Welch for Melody Maker , the article being dated 12 October 1968 , he says that LZ were due to go to the US in early November of '68 so perhaps the first US tour dates weren't firmed up at the time.

There are numerous attempts at the Heartbreaker guitar solo takes as evidenced on the Led Zeppelin - Led Zeppelin II Classic Records 45rpm Treasured Version (Empress Valley Supreme Disc - EVSD-953), 1CD [2017] boot, with Page saying "I'll do another couple" at the beginning of the available tape, which has several cuts. Definitely a day when Jimmy lost the thread several times as he tuts to himself throughout the session.