Answering my own question, in case anyone has this issue in future.
I sent a query to the UK tax authority (HMRC) who responded as follows:
"As you are providing supplies to a UK business from outside the UK, your client would be liable to account for the VAT using the reverse charge procedure. This is explained in section 5 of VAT Notice 741A: place of supply. Therefore, you do not need to charge or register for VAT, however you may choose to register for VAT in the UK if you wanted to. This is explained in VAT Notice: 700/1 Should I be registered for VAT?"
Basically, there are different requirements depending on whether you are supplying a business or consumer in the UK. If you are supplying a business, you do not have to charge VAT. If you are supplying an end consumer, you would be responsible for charging VAT in some circumstances. Hope this helps.