You will be able to get lounge access at LAX, TPE and SIN based on being *G (granted to Premier Gold and above), regardless of PE or regular economy. In SIN, you'll have access to the KrisFlyer Gold lounge. It doesn't look like there's anything at HKG besides a United Club, which is temporarily closed, according to the Star Alliance lounge finder tool. It's been a long time since I've flown through HKG, so not sure if others can weigh in on other potential options.

ETA: UA's status match page does have this in the fine print: "Premier membership materials will not be issued for the 120-day offer period. You may enjoy Premier benefits on United- or United Express-operated flights by providing your account number when making reservations or at check-in. During the 120-day offer period, you may not have access to all reciprocal Premier benefits with United's partner airlines, including access to Star AllianceTM lounges." I've never done the status match so can't weigh in on whether or not you'll have access to *A lounges. At the very least, you'll have access to the UC at LAX.