As far as June goes, expect the same weather there as in Miami: hot, humid, rainy, and a few hurricanes.
As far as Cuba goes, I went a few years back on a trip to visit where my family grew up, went to school, etc. [As my mother is Cuban-American, we were legally granted permission to visit by the U.S. Gov't.]
We went to Havana, Havana "Vieja" (Old Part of Town), and Varadero.
Varadero was an hour or two away from Havana and was filled with German and other European tourists (like you!). It was alright, but I wouldn't go again...I do live in Miami and a beach like that isn't all that great.
Havana was really fun as I expected it to be totally different then what it was like. Note that the food there was, and I am sure is, HORRIBLE. Oh wait, your from the UK

. Honestly, really bad food. At least coming from someone who loves Cuban food.
I did not feel unsafe while there and had a few chances to distribute toiletries from the hotel to people we meet who needed them desperatly. The island's people seemed happy, but the conditions we saw in some areas were a notch below deplorable.
Overall, it was an amazing trip for me and my family because it was the first time many had been back to the island since leaving ~35 years ago. My mother and grandmother's home were still intact, and basically nothing much changed.
What area of Cuba is Cayo Coco? Is that a beach resort or in Havana?