Brian is watching MTV (which debuted in August 1, 1981) and later Michael announces that he's been invited out of town for Memorial Day which was May 25, 1981.
When the Royal Wedding is shown on television it would have been the middle of the night in both California and Alaska. The wedding was at 11:20 a.m. in London which would have made it approximately 2:30 a.m. in California and 1:30 a.m. in Alaska. California is 8 hours behind and Alaska is 9 hours behind London time. However, the wedding wasn't necessarily being shown live and recorded highlights from the wedding were shown by US networks.
The song "Do You Wanna Funk" by Sylvester James is heard, but this song was not released until 1982 and this miniseries is set in 1981 (curiously, the movie Longtime Companion (1989) makes exactly the same mistake during its opening scenes also set in summer 1981).