I did my daughters 16-17 railcard last weekend. Used her passport details and went through instantly. Only hick up was the activation code was sent to her email instead of mine, but we found it pretty quickly.
When I renewed my disabled card last year, it had lapsed about a month after expiry so I couldn't auto renew it and had to submit evidence again. Phoned them up and and the call handler told me to email him a copy of bank statement blanking out everything except the PIP credit and a copy of the out of date award letter with matching NI to the bank credit and took my payment for the card. Even after all that, my activation code arrived about 3 minutes after I emailed the documents, so even complicated ones can be very quick.