Do I need a pair of rain boots on my trip to England?

I live here and I don't own rain boots, or wellies (Wellington Boots) as we call them.

If it does rain super heavy, which is rare, it tends not to last too long.

Not necessary. I travel to Lndon most often in November when there's a better chance of rain and I've never taken rain boots. I take two pairs of shoes. KEENS. Leather slip ons and the low ankle Voyager Hiking shoe. Both lightweight yet durable and because I walk a lot both extremely comfortable. If I traveled in June like you are I'd substitute my Keen sandals for my hiking boot.

only if you're walking dogs in muddy fields or going to Glastonbury

When ramblin'on writes Glastonbury, he means the music festival, which is notorious for wet weather. In the town itself, ordinary shoes are fine.

I can loan you my very large, heavy muck boots, but I have never taken 'em on any international trip. Nor do I own a pair of waterproof shoes. Worry less, travel more.

You should not need the Wellingtons that Compo Simmonite (Last of the Summer Wine) tromps around with but may be best served having waterproof lined (e.g Goretex or similar breathable but waterproof membrane) shoes/boots. With almost three weeks, the probability of at least a day or three of precipitation is pretty high. That said, if you are crossing bogs in the moors, Wellies may be appropriate if you can keep them from getting sucked off.

Compo in his Wellies: https://greensleevestoaground.files.wordpress.com/2010/02/last-of-the-summer-wine-bill-owen-and-william-compo-simmonite.jpg

Wellies are only worn for walks in the countryside, not in cities. You won't need them in June anyway.

RE: Wellies are only worn for walks in the countryside...

I would insert: "only SHORT walks...." Wellies do not have much support and tend to be loose fitting. Even if sized snug, they are pretty flexible and will not offer much support. The would be great for mucking out cow stalls but are not suitable for distance walking or tracks requiring boots with ankle supports. Note that Wellies don't "breath".

PS You didn't say where in England. Here is the UK Met climate page:
http://www.metoffice.gov.uk/climate

Regional climate data:
http://www.metoffice.gov.uk/climate/uk/regional-climates

For example, SW England rainfall:
http://www.metoffice.gov.uk/climate/uk/regional-climates/sw

The months from April to July are the driest when the sea is relatively cool and the Azores high pressure system exerts more influence.

No, it really doesn't rain as much as people try to make people think it does. On the last trip was there the month of September and a few days in October, only had two hours of rain the whole trip. Have been there many times during the summer only a couple of days had rain, and it didn't last long.

I love my waterproof Ahnu hiking shoes. You never worry about rain boots, and they are so comfortable.

Hi! I have been living in London for two months and I didn't get used to living with such unpredictable weather. So I bought a pair of rain boots Kamik http://www.comparaboo.com/p/kamik-women-s-heidi-rain-boot-yellow-9-m-us?id=194472&list_id=715777. I know they're not the typical rain boots you'd see but I think they're cute. (I don't have enough energy to get up and take a picture of the real things so this pic from site will have to do lol).

I bought these for my UK trip last August: http://www.zappos.com/merrell-moab-mid-waterproof-grey-periwinkle?ef_id=VhLCugAABWhblc4p:20160518164117:s

I wore them to Stonehenge (where it rained) and while touring London (where it drizzled). They supported my weak ankles and provided superb traction on slippery cobblestones and wet Tube station floors. My feet stayed dry but never felt hot. I never got one blister during my trip.