How much of a maths background is necessary for a PhD in Software Engineering?
It is a bit hard to say, because whilst you say you don't "know" math, you must know - some - from your Sofware Engineering undergrad and Comp. Sci. pre-university education. My background is AI, a very wide field that can involve all kinds of mathematical formalisms - from Tarskian Semantics to Calculus and most researchers aren't an expert in all of the areas AI makes use of, some are completely unknown to some researchers (in my experience). So your question, "do I need to know maths", depends on what you want to do - you don't need to know ALL of our current mathematics (no-one knows that anymore in any case).

So my first suggestion is to read some Software Engineering papers and see how much mathematics they use and what kind. If you want to do quantitative research you will need to know statistics. If you want to develop formalisms to support Software Engineers you will need to know some logic and basic ways to prove theorems, and so on.

My second suggestion is to think carefully about the Masters you do in order to encompass the areas you want to learn about. But, it is perfectly possible to learn much of this on your own - at first it'll seem difficult, of course.

To try and give you some encouragement, maths is often Greek to people, because of all the Greek letters, but in my view it is also possible to pick up the necessary parts. I taught (during my PhD) students coming from a wide range of backgrounds some elementary formal logic and Artificial Intelligence techniques, most of them picked it up fine and I think from there they would have been able to pursue a PhD in AI - I have also met people coming from non AI, Comp. Sci. or similar backgrounds that have pursued a PhD in AI.