The easiest solution would be to get married. And if you want to work then you would need a Dependant Visa (which requires you to be married) or be sponsored in your own right for an Employment Visa. However, for clearly established relationships then some people have reported success in getting Extended Visit Visas for unmarried partners, which do not allow you to work. As a British citizen you would get 6 months per visit as a visitor anyway, although after one or two back-to-back renewals of that you might start to have some questions from immigration. I'm not exactly sure how visas for the kids work, but I guess they follow from the mother's.
On the basis of what you have told us, my advice would be to look very seriously at getting married - the process is absolutely trivial in HK - at its simplest, you have to give 2 weeks notice and then go to a solicitor's office (or anywhere else you choose) with 2 witnesses and do a 10 minute ceremony and form signing. Costs less than HK$4000 all in. If you prefer, no-one except the solicitor, the witnesses and Immigration need know that you are married (no requirement to change names or anything) and you can do a "public wedding" (with no legal significance, but the one that "counts" socially) any time you want in the future (or not).