Can child travel to UK on tourist visa while MN1 3(2) citizenship application is being processed?

We need to take our foreign-born child to the UK on her South African passport while her MN1 3(2) citizenship application is being processed. Can we get her the normal tourist visa using her South African passport and take her in on that?
If so, could we apply for her UK passport while she was there on a tourist visa, or would we have to bring her back to SA and do that from here?

With thanks

I think that it should be possible for both.