I've never been charged a phone service fee as a GCH, and most of the time they explain that there's no fee because I'm gold.

I have an email from Customer Services in April that includes the phrase:

as a Gold card holder the £15 service fee will still remained waived when making changes via the contact centre.

So I'm with angatol on this. Unless BA have done another stealth change, I believe GCH do not pay the phone service fee.