I like tea, so I can't comment on something that doesn't taste like it. My understand has always been, though, that it's not the tea, it's the Honey that soothes.
Most advice is to use a salt water gargle for a sore throat. A few things to consider with a hot liquid - the heat (if not cool enough) may actually aggravate the inflamed area and make it worse. The citric acid in lemon may do the same. Honey is not pasteurized (which is why it's not advisable to give to young children), so it may contain bacteria which could exacerbate a sore throat.
Personally, I'd call the pediatrician/family physician and ask if it's better to use heat or a cool drink to soothe your throats during your cold.
Good luck! Hope you all feel better soon.