Works fine for me, however, just wondering why you are using a web browser on a phone instead of a mail app.
OK, but much simpler to configure a mail app such as Blue Mail to collect it directly rather than forwarding to Gmail.
Why don't you add your BT email account to the gmail app and it will fetch your BT email as per the gmail app's normal polling time or every time you open the app.
Gmail is configured to collect my email from BT. Gmail doesn't check for new mail at BT when it is started. Google don't specify a collection schedule only that it will be done periodically. If your waiting for an email it's quicker to go directly to the BT Web Mail. But BTs Web Mail doesn't work on my phone in Chrome. It didn't work in Chrome on my last phone either.
I suspect some setting in Chrome is the issue.
What you have done is to set webmail via your gmail account to fetch your email from BT and it may well behave as you have said however the gmail app is just an email client. If you use the gmail app and add your BT email account to the app it will access your BT email account directly in the same manner as any email client would.
I can access BT email on Chrome so I doubt it is a Chrome issue as such but if Chrome is not doing and has not done what you want, stop using it and use another browser.