Unlock someone else's health code bound to my mobile phone. Can I still get the text message?

I can't. Because the health code is in your WeChat or Alipay.

Binding is equivalent to getting real-name information from your WeChat or Alipay, instead of getting your information from a mobile phone. When you don't log in to your WeChat or Alipay on a mobile phone, your health code information is naturally not on that mobile phone.