I just wanted to comment that I think waiting a month or so to adopt a new kitty is a good idea.
First though, I just wanted to say I am so sorry about you being blindsighted at the vet. I had a similar experience with my cat. I noticed she was holding her mouth open slightly, my boyfriend thought I was seeing things. I took her in, thinking it would be a minor dental issue. My world stopped when the vet said it was most likely fatal jaw cancer. We only had about 4 weeks with her after the diagnosis, I couldn’t imagine how horrible it was to have to put your kitty down that same day. I want to say with as much kindness possible, you did the right thing, and thank you for putting your cat’s suffering above your own emotions.
To answer your main question though.
I grew up in a household where shortly after an animal passed away (mainly rats, gerbils, fish, and hamsters, but we also had a kitten die from a fatal illness) my family would get another animal only a couple days or maybe a week later, sometime the next day. It wasn’t because we wanted to replace them, but more we were ready to add a new friend to our family.
I never had to deal with the feeling loss after the death of a pet, as we had a new one to fill our hearts and to love on.
As an adult, after I had to put my kitty to sleep, I was a massive wreck. I had raised her from a tiny kitten to a 10 year adult cat, and her loss destroyed me. I wanted so desperately to get another kitty to take the pain away but couldn’t because for various reasons. I stlll break down sometimes when I think about her and how much I miss her. It has been over two years, and the pain is still sharp. It wasn’t until a year after when I found a stray pet mouse who I adopted did the pain dull a little bit when I thought about her.
All that said, I highly believe not having to process grief from a pet’s loss as a child is why I have had such a hard time with her loss now as an adult.
Waiting a bit, while it is hard now, I think will be a better option for her in the long run in learning how to cope emotionally.