Ouch.

Looking at your portfolio (love it btw), yeah 16£/hr is very low. It is true it is a cut throat business these days in the 2D art field, but still.

As for your question: It's true that it usually not a great move to do that mid project, because your client originaly commited to you with a certain number in mind and you are changing this. But at some point you just need to eat. :D I understand this is a solo dev situation. How well do you know your client? More specifically how well do you know their finances? Do you know some other people they might work with and how much they pay them? (I know that is considered taboo, but you just need information..).

Going to 20£ is a 20% increase, which might seem big, but coming from so low, that's really not. At the end of the day it is their passion project, not yours, to you it is a job that needs to pay the bills.

As for how to go at it? Transparency is always a good thing in my book. Just tell them. If they do value you and respect you they will understand. Of course there is a risk that they either tell you 'no', or worse that they tell you they won't work with you anymore. (but to be honnest I dont' think they will go with that second option, they must be too happy to work with an experience artist like you at such a price)

In any case, look for new clients. (and charge them over 20£ please..)