When you say not too hot have you got a range in mind?
A lot of the Med will be properly hot late July - August (Average temp low to mid 30s). Would that be too hot?
If you’re wanting to stick to the Med but not wanting it too hot, you could look at the South of France, parts of Croatia, Northern Italy, Corsica, The Balearics, The Algarve (although they can still be pretty warm!)
You could have a look at:
Zadar (Croatia) from Bournemouth (beautiful city, options for beach holidays close by, stunning coastline)
Carcassone (France) from Bournemouth (medieval history/architecture, mountains, rivers/lakes, nature, options to do a Eurocamp style holiday)
Bergerac (France) from Bournemouth (medieval history, wine, great food, great landscapes, options to do a Eurocamp style holiday)
Faro (The Algarve) from Bournemouth (lots of options for beach holidays, big resorts to smaller towns, great beaches, attractions for kids, history and nature)