I checked St. Regis website again and found that the Brasserie has weekday lunch from 12 to 3pm and high tea from 3pm to 5pm, hence it's unlikely they can sit us down at 1pm, so our best bet is to sit at cozy corner in lobby until 3pm. As for Goodwood Park, this is what's on their website:
English Afternoon Tea Buffet
Available from 2pm to 5.30pm (Monday to Thursday),
1st seating: 12pm to 2.30pm; 2nd seating: 3pm to 5.30pm (Friday to Sunday and Public Holiday)
Maybe that's partly why the Facebook group picked Wed 3 Dec (instead of Friday) to go to Goodwood Park so at least they can sit from 2pm onwards.
So in conclusion, we haven't found a high tea place (with wifi) that can let us sit from 1 - 5pm.
Do you want to go back to check the timings of the other hotels that don't offer free wifi, e.g. Regent, Mandarin Oriental etc.? Then I'll make a decision whether to take half day leave.