Photo required for two together rail card

You do not need a photo for a senior railcard.
You do need photos for the TwoTogther card but whilst they need to be the same size as passport they do not need to be the same quality.
You can buy railcards in advance online to be carried on your mobile phone if you wish.

You can get information directly from this source.
https://www.nationalrail.co.uk/tickets-railcards-and-offers/railcards/national-railcards/

I recently purchased the electronic version of a rail pass and it was quite easy to do.

Specifically-
https://www.twotogether-railcard.co.uk/

and

https://www.senior-railcard.co.uk/

The app version of the senior Railcard requires a photo but then again it is easy to take one with the phone!

There was a period where the online system for a plastic card was requiring a picture but that was a software fault that I daresay has been fixed.

a plastic Senior Railcard doesn't require a photo, and they are easy to get at a staffed rail station.

The website says you need a password size photo of each person for the Two Together Railcard.

As I said above you need passport sized photos for the TwoTogether card but they do not have to be of passport quality or meet passport regulations.

Much easier using the app - and having done it a number of times, aside from size the pictures are most definitely NOT passport quality

Can you just take cell phone pics of each other and present them to the station where you get the Two Together Rail Card? Or do you need something printed out? My husband and I are planning to get this Rail Card once we fly into London in early September.

You can take cell phone selfies, but they need to be printed out as a roughly passport sized image, without being pedantic on the exact measurements.
A station can't handle jpegs or whatever format they are in.