a few disconnected thoughts -
the hosting and domain name is relatively easy.
some providers offer free or discounted hosting to charities. some free hosting is flakey, or propped up with adverts (that may not match the message you're giving) or be very restrictive in the amount of bandwidth you get (which may be ok over the course of a year, but might be difficult if you have seasonal peaks of people wanting to look at the site, e.g. if you do big events, or a specific campaign, a few times a year.)
charity i used to do the website for used
34sp who have a free charity hosting scheme - you need to be a registered UK charity, it's not guaranteed for life, but i must have set the thing up about 10 years ago and as far as i know it's still going (i parted company with said charity a couple of years back) - the agreement needs to be renewed every year, but involved a fairly simple exchange of e-mails. i found them good in terms of what it did, never aware of any problems / downtime, and they were good with technical help when i was moving it to them (it had previously been hosted through someone's business as a favour, but for various reasons that had to end.)
the domain name (as in the right to 'own' charityname.org.uk or whatever) does have to be paid for, but this is a few quid every couple of years.
moving a website and e-mails from one hosting provider to another shouldn't be that big a deal.
not sure about costs of doing it all professionally - depending on what you do, the design stuff could range from not very much to a heck of a lot. but not sure where to start coming up with an outline specification / budget / full specification.
probably worth thinking about how you're going to maintain / update it in the medium to long term as well as the one-off project to set it up. websites that clearly haven't been updated for years (e.g. having details of 'future events' which are now in the distant past) don't inspire confidence.
this can be difficult in voluntary organisations - you don't want too many people doing the website as you can end up with confusion about who's doing what, but having a one person department can also be difficult if that person leaves, has work or family issues that means they are too busy to do it, or has a hissy fit and falls out with the committee - in my case i continued to maintain the basics on the website until i could hand over, then managed the hand-over, but i've known some organisations where that doesn't quite happen...
to some extent, you can get round this by website being relatively static and putting latest news / events out via social media (although worth making sure that there's some sort of overall control over communications - if you end up with one person doing a newsletter, someone else doing a website, someone else doing social media, and them not talking to each other, or being given conflicting information / instructions by different committee members, that's not good either)
i'm not sure this answers much of your question, and no i'm not volunteering - the website i did was on technology from the last century, and keeping it going in to the 2020s was an increasing struggle, but getting committee approval to do something about upgrading it (even at no cost to the organisation) proved an even bigger one.
