For Canadian applicant, no. You should get email notifications (not always), and if the wait exceeds a month or so (for non-priority), you can inquire through https://www.gov.uk/contact-ukvi-outside-uk. Use email.
@joppa I have a similar issue. I have made my FLR(M) application. Made it on April 10th, found out about the NHS fee thing after the fact, so had to pay an extra 500 plus the 48 for the application fee. Sent that stuff out not last weekend, but the weekend before. The 48 is just now being taken out of my husbands account. Im not sure if this means theyve just now started looking at it again, or are done looking at it. So I called, used the "check progress of application" option, offered to give a reference number, and was told that they cannot check the system for the progress of it. I am confused as to what 'check progress of application' means if it means I cannot call to check the progress. Any insight? Husband thinks I should call back and speak with another agent. Not sure tho.
Perhaps. He did say something about the time not exceeding the 8 weeks or whatever. They should change their call headings then..."check progress of application if after the allotted time" sounds better. Lol anyway, thanks!