Only the wording of the warranty can help you. Some warranties allow transfers like a cars for example, others don't and state non transferable. As you werent a party to the original contract, or as a named third party under the rights of third party act you have no rights under consumer law.

I know you have to keep your Costco membership open or it voids your warranty, so I'm guessing that it probably wouldn't be transferable

Here is the extended warranty terms from Costco online. I think the clause D applies to you. So if not a member, it might not transfer. Having said that I find costco customer care is brilliant. I got a faulty CCTV camera which they return with full money back some 6 months after purchase, I had lost my reciept as well, they just found details using my membership details. I would recommend becoming a member if you can.

** Extended Warranty Terms & Conditions
See Manufacturer’s warranty for specific coverage terms. For warranty service contact Costco Concierge Services on Freephone 0800 066 5091.
This warranty is subject to the terms and conditions detailed below in paragraphs (A through F) but does not affect your Statutory Rights provided by law in the United Kingdom.
In the unlikely event of a malfunction occurring, you will receive free of charge service covering both parts and labour.
Repairs will be carried out by either manufacturer, NESN (which is a national service organisation) or another Service Centre approved by Costco.
This Warranty does not cover accidental damage, misuse, computer viruses, software, apps or any fault or defect caused by fire, lightning strike, water or sand ingress, neglect, misaligned or unsuitable aerial, inadequate internet connection/Wifi signal or fair wear and tear. In addition, any attempt to repair the product by non authorised personnel will invalidate the Warranty.
Consumable items are not covered under the Warranty. They include, but are not limited to, batteries, bulbs, bags, cassettes, SD cards, cables etc.
Warranty applies to Costco UK members and costco.co.uk purchases only.
This Warranty applies to mainland United Kingdom only.
In order to provide you with Costco Concierge Services, Costco may share information about your electronics purchases, such as make and model, with third-party providers. This will not be used for Marketing purposes.

No it has to be a Costco Card Holder that deals with any warranty claims, not transferable to 3rdparty. Costco Extended warranty has always been that way. Even as the original purchaser if you end your card membership the warranty expires.

They are generally pretty good at Costco, although if you aren't yourself a member it might make things complicated, you would get more mileage saying it was a gift from a friend or just getting him to help directly.