Honestly, I would say ask her, not just her mum. She may feel completely different than her mum did.

Have you asked your partner "how can I support you?" "do you want to talk about it or would you prefer we talk about other things?" When I had thyroid cancer (I know that's not really the same), one of the best reactions I got was from the head of department who just asked me what I wanted and how she could be supportive?

I do think your partner's mum has a point about talking to other people about how it affects you. There is a principle that you should "dump out," which means you get support from people further from the issue than you. This doesn't mean you can't talk to your partner about how it upsets you, but you should be spending more time listening to how it affects her and reacting however she wants you to, whether that be by just listening or by being positive or by changing the topic to other things or by taking her worries seriously or whatever (I'm sure I can't even imagine all the possible reactions).

Have you friends or family of your own you can talk to? While I think it's great that you and her mum were trying to support each other, as you are in a similar position, needing to support her while probably being devestated yourselves, I also think it would be good for you to have somebody closer you can talk to. Somebody who is closer to you than to your partner, I mean, and whose main concern is going to be how it affects you.

I think though, that you need to talk to your partner and let her know that you want to be with her but don't know how to support her. Tell her you are sorry you backed away and ask how you can support her in the future.

Ask her rather than her mum. I know her mum knows her well, but...given that she also went through cancer, she may be giving you information based on what she wanted when she had cancer rather than what her daughter wants.