Hi!  Great questions.  Looking at the geology map for the Isle of Wight, the rocks are sedimentary.https://en.wikipedia.org/wiki/Geology_of_the_Isle_of_Wight
The rocks you collected on the beach appear to be well rounded vesicular basalt.  Searching the British Geological Survey website, I found Permian (very old rocks) described as extrusive lavas.  http://mapapps.bgs.ac.uk/geologyofbritain/home.html?
They are due west of the Isle of Wight, west of Dittisham.  Continental glaciers scoured across England.  Erosion and wave action could account for the heavy hard basalt cobbles present on your beach.  Geology puzzles are amazing!  If I had not found basalt near by with a reasonable transport explanation, then I might have thought about meteorites.  Vesicular basalt is formed as the gas bubbles travel up through molten lava.  The holes are the evidence of gas presence.  The vesicles are sometime filled with other minerals at a later time.  Sharp eyes!!  Good work!!  

Hi again, thanks for the answers. Do you know what the (as you put it) minerals are and why they grow in water? Can you tell how much they're worth?

Rocks are made of minerals.  Chemical elements make up the minerals.  The earth is complex and geologists are still learning more about it.  Deep flowing groundwater flows from the hot rocks to the cooler rocks.https://en.wikipedia.org/wiki/Hydrothermal_circulation  Hot springs and geysers are a surface expression of these waters.  The water carries minerals which can be deposited through precipitation along fractures or fill air pockets.  Hardwater deposits (calcium) is an example we often see on faucets.  This experiment with sugar is way to actually grow your own crystals.  https://www.kidzworld.com/article/26598-make-your-own-crystals  In the case of the basalt, the mineral that is generally deposited is a form of zeolite. https://en.wikipedia.org/wiki/Zeolite  Sometimes the mineral is a silicate called quartz.  Have fun learning to grow crystals and learning more about the minerals and the rocks they form.  As far as worth, these are very common minerals with little value.  I suggest researching mining and the rich history around mines in Great Britain to learn more about value.  Have fun!

Another question! Do you know what zeoliye mineral this is because I found A) It is some sort of product, and B) There are different types of zeolite (according to Wikipedia).

No, I don't know what the zeolite is.  There are a lot of zeolite varieties.  It also may be a quartz infilling.  In order to identify zeolites, it takes further analysis with a petrographic microscope and geochemistry.  You are asking all the right questions.  From here, I suggest talking to a local geologist and seeing what they have to say about the types of rocks you found and what their geologic history is.  Besides in universities, geologists work for soil conservation districts, as engineering geologists, British Geological Survey, possibly with a forestry service company, with mining.  At the universities, sometimes geology programs are called earth science.  Best wishes on your next line of questions!!!

I thought of another question! Is it very rare to find Permian Vesicular Basalt? If so, how rare?

Hi!  Bedrock is the foundation.  As geologists learn more about the earth, we're gaining a better understanding of all the unique connections between hydrology, soils and geology and how humans interact with the environment.  Volcanic processes have been a key component through the earth's 4.5 billion years of life.  Rocks are categorized into three main areas:  sedimentary, igneous and metamorphic.  Basalt is igneous.  So to answer your question, Permian basalt, although rare in your corner of the world, is not rare on earth.  If you are interested in further reading, I recommend Simon Winchester's "The Map that Changed the World" about a ditch digger in England.  Robert Thorson's "Stone by Stone" and "Exploring Stone Walls: A Field Guide to New England's Stone Walls" explore the connections between England and New England.  I'm sure you can find similar books written about the geology for England.  <Grin>  Once you open up the door to learning about geology, it's a wonderland of puzzles.  Rocks tell stories.  We just need to listen.  Have fun!!!