Can't you just ask her when her last period was? Say you wanted to work out the due date using an online due date calculator so you need to know the last menstrual period to do that. She will know this date as the doctor will ask her.
I am an amateur expert on home pregnancy tests. The very very earliest you can really get a very faint positive is 9dpo, first response and perhaps superdrug may, if you are lucky, give you a faint line then. The clearblue with conception indicator do not really begin working until 11 dpo and even then, that is if you are lucky! Of course, if you get 1-2 you could be anywhere from 11dpo (days past ovulation) to 15dpo. Or even a couple of days more as the tests are not an exact science.
I think this conception would have resulted from having sex before the third of the month. Clearblue say you can use the test 4 days before your period is due, so that is 10-11 dpo. Assuming ovulation took place on the third, that would mean no positive result would be possible till approximately the 13th or the 14th of the month.
However, sperm can live for a few days in the fallopian tubes. Lets say she ovulated on the first of the month (quite possible with a first positive test on the 11th). You could have had sex five days before that and it still could result in a conception.
Another possibility is you had a shag on the 21st before you went away. The spermies lived for, lets say, 5 days. That means a conception could have taken place as late as the 26th or even the 27th October if you have some long lifers. Did you use contraception before you went away?
Add 2 weeks to that and you get the 10th November. The clearblue test does not show up early for everyone.
It is impossible to deduce paternity from a home pregnancy test. However, when you get a scan, take 14 days off the dating they give you, count back on a calendar and that will give you an approximate date of conception.