In your situation I'd say you have been offered more work and your rate is going up to £15 an hour - that's still very low but the problem is you've started off low too so proportionally it's going to be a big jump. I'd be interested in what others think but I wouldn't recommend you say ok my rate is now £25/£30 an hour, it's too much of a leap.
I agree that in an ideal world you’d have started at £25 an hour minimum, but that’s not possible to achieve with this client because of your historic rate. I’d tell them that you’ve been assessing your business rates and comparable work based on the other freelance commitments you have and your minimum charge from X date (next month) is £18p/hr. However as they’re such a long-standing client, and a charity, and you enjoy working with them, you are willing to discount this rate to £15p/hr.
If they come back and say that’s too much, ideally wait for them to counteroffer. If they don’t immediately, ask them to - ‘Thanks for letting me know, I appreciate budgets are squeezed for everyone at the moment. What budget do you have for this sort of work - perhaps there is an agreement that suits us both?’
Then you can open yourself up to negotiating a lower rate for guaranteed regular hours, or a short-term rate of e.g. £12.50 to be reviewed in January, or whatever. That’s all if you don’t want to lose the work so don’t want to be totally hardball.
But they will know they’re taking the piss unless they’re absolutely tiny and work on a skeleton staff, which it doesn’t sound is the case.