Like it Kid 😃

One - A website AND a forum. If it were me get a domain and web hosting from anywhere, I've used Falcoda for ages and find them pretty cheap and quick with support. If cost is an issue, you can't go far wrong getting Wordpress installed on there. There are a gazillion themes with the better ones only costing a few quid (they also give you more control on changing the theme compared to the free ones). Then on your hosting setup a sub domain (takes seconds and is ready instantly), call it forum.oneadaywebsite.co.uk or summat. Then there are lots and lots of providers of Forums, this one is done by Sparklit and is a pretty cheap monthly fee, it does get more costly the more views there are per month (we're right on the edge of the lowest price band, if popularity keeps increasing its gonna cost me more!). Then when you setup your forum, do so giving it the website sub domain you made earlier, then on one of your menus, links or whatever simply link to your forum.

I don't see the sense of spending hundreds and even thousands of pounds to have this bespoke built for you (unless you're hell bent on it of course), there's too much good stuff out there off the shelf. Web development is a dying trade!