I’ve managed to extract the direct link to the form that comes up when you click the blue link:
https://omni.eckoh.uk/v03.5/providers/HMPO/window/windowChat.html?t=HMPO-08f5bf1501cd534d8d947263d4b307ff&p=HMPO&pid=2&d=717&n=&e=&c=HMPO-144b74ed009a7e2517d&tel=&ch=CH&iid=STC&psk=chat_a1&m=3&fcl=0&r=Chat&srbp=&s=https://omni.eckoh.uk/v03.5&surl=https://omni.eckoh.uk/v03.5&iif=&l=en&auto=&ud=&u=https%3A%2F%2Fwww.gov.uk%2F&wo=&uh=2022-4-28%252007%253A28%2Fhttps%3A%2F%2Fwww.gov.uk%2F&wait=
This is pretty much the blue link itself, so you don’t have to wait for the “All our agents are busy” message to disappear for the blue link to show. Hope it helps a tiny bit.