The weather in England in June (much like the rest of the year) is very unpredictable, so you will want to be prepared for anything. The average high is 68 F and average low is 55 F, but that does not mean that we could not also see some much colder or hotter weather. On average there is 11 days of rain (but it is common in London to have a brief passing shower and the rest of the day cloudy or even sunny). In all likelihood, the weather in London in mid-June will be similar to what we are currently experiencing in late April and early May here in Maryland (this year has been cooler and wetter than usual).
So, clothing recommendations? A light water-proof jacket is a must (and a small umbrella is not a bad idea either). You may be able to wear shorts on a few days, but bring a pair of jeans for those cooler days and evenings. A combination of short sleeve and long sleeve shirts is a good idea. Bring something nice for the nicer dinners and our theatre performances. And, most important, bring some comfortable walking shoes.
See the weather widget at the bottom of the page to keep an eye on the weather between now and the time we leave. You should also check the 10 day forecast for London before leaving.