Chris, I'm probably like you, but I don't have a partner, although at the time, both parents were still alive, and I'm close to my brother's family.
I also didn't tell anyone while I was going through diagnosis (which in my case took 6 months). I could cope with the appointments myself, but I didn't have the emotional reserve to support family members, and when you tell someone else, you do have to support them - it's probably even more a shock to them than it was to you. I wanted to be able to support them when I told them, which included being able to answer questions such as my prognosis, treatment, etc. so I needed to know my diagnosis first, which was taking some time. And I needed the emotional reserve to support them.
When I thought I had my final diagnosis, I told my brother, in part to discuss how to tell our parents. It was quite a weight off me to have told someone, and he wouldn't have pushed to go to appointments anyway, unless I asked him to, and I didn't really want anyone else at appointments anyway. I told my parents, and they were actually too shell-shocked to ask me anything. In some ways, I would have preferred not to tell them, but I was partly their carer, so it would have been impossible to hide.
Then the hospital changed their minds and decided they still didn't understand my diagnosis, so I had more consultations and tests. Just as I walked out the first of these, my phone rang, and dad asked what had happened. I hadn't yet processed the information myself, and I couldn't handle being asked about it yet. That absolutely confirmed in my mind that it was right for me to handle this on my own. For subsequent consultations, I said I was just going for a blood test, so this didn't arise.
Everyone is different though. Most people will want a partner to help support them, although I do know of some other people whose partners are not able to do that, and the patient ends up having to support their partner in the consultations, instead of trying to get the information they want from the consultant. I come across this also in the 1-2-1 support I do now, and there are things guys might want to ask about without their loved ones being in the room.
It's good your daughter feels comfortable with you helping her and that must be a big comfort, although that may be difficult for your wife to understand.