So it looks like Two Together Railcard is our answer. Do we need to take passport photos with us, or can we take a photo with our cellphones? We will buy the card at the station in London before we catch our train. Thank you!
2018 we needed passport photos for our Two Together card. We got it at Paddington Station, took about 10 minutes.
I brought the photos with me and completed the application form in advance. It made buying the pass very easy. I haven't read your other thread so I just wanted to make sure you knew you could purchase your tickets in advance stating you have a railcard just as long as you purchase one before you actually travel.
Whilst you need passport sized photos they do not need to be of passport quality with the various rules that countries insist you have. For example here in the Uk an adult can't smile on the passport photo. No such restrictions for the railcard.
I had some extra passport photos but didn't use them. I photocopied one at the correct size, on regular (cheap) copier paper.