Globe theater tour/where to go for Afternoon tea with kids

The tour is first self-guided in their museum and then you meet a tour guide downstairs who tells you about the theater as you tour it.
Depending on the ages of your children, the museum should be interesting for them. There is a history of the theater and then lots of costumes and props from various productions. They have interactive exhibits for kids where they can push buttons, slide doors open, etc., so they should enjoy it. The tour guides I have had, have been entertaining and tell fun and interesting stories from all periods of the theater's history.
If you want to go see a play but aren't sure the kids will last, get groundling tickets and stand in the main space in front of the stage. If you need to leave you can, you haven't spent much money, and the kids still will have the experience of seeing a Shakespeare play in the setting for which it was intended.
If you do go to a play, definitely get tickets online, in advance, because they do sell out.

I have taken the tour and loved it. My husband didn't want to go, so I dumped him at the adjacent Swan Restaurant where he hung out and read while I was on the tour.

The Swan has a variety of things on its menu, but it does have a special afternoon tea. This is the link to the menu -- http://www.loveswan.co.uk/perch/resources/afternoon-tea-1.pdf. Be sure to go to the 2nd page of the pdf for times, price and reservation info.

The tour is not available when there are plays being performed, so you will have to go in the mornings or sometimes in the brief gap between performances. Sometimes the tour instead goes to see the remains of the Rose Theatre, a genuine early modern theatre whose foundations are preserved in the basement of a nearby office building.

There's nowhere nearby that does "afternoon tea" in the Ritz sense of the word. It would be worth visiting Borough Market if you are on a day that it's open, although on Saturday lunchtimes and afternoons it is intolerably busy.