I have been a part of Canadian graduate student societies in the past, and I am not familiar with the culture/formalities of forming societies in Britain. I have been the Vice President (and acting President) of a 600 member society (pretty small) so most of my knowledge comes from personal experience.
Forming a society is actually quite straightforward, though a very involved process (it will not be easy). It's quite similar to starting a business. Essentially a Society is just a collection of like-minded individuals. Anyone in a free society can form a Society whenever they want, and have as few or as many members as they want, as long as they abide by their country's laws of course!
Zeroth Step: Maybe join the existing Society?
But before I get to the answer about how to found a society, have you considered running for Executive positions in the British Mycological Society? They would already be set up with resources, members, bylaws, etc. so it would probably be easier to join them and take up a leadership position to guide the society in the preferred direction than to create a brand-new society. Just a thought.
First Step: Learn about Societies
The first thing that you will want to do is take a quick look at Robert's Rules of Order, especially Section 68-70 in the Fourth Ed. You can find the fourth edition online in the link that I provided. If you want the latest edition (11th) you will have to buy it. I highly recommend this book and it has served me well in my society experience. The 11th edition has a tonne of info about how to found a society, how to write bylaws, how to track finances, how to hold elections and (most famously) how to hold meetings. The fourth edition doesn't have as much info about forming a society but it is still quite useful (and free!). It is written from a somewhat American perspective but I should think it mostly carries over.
You'll want to familiarize yourself with how similar societies are operated, what their constitutions/bylaws look like, etc. This shouldn't be too hard: you can probably check out the existing British Mycological Society.
Second Step: Find your supporters
Robert's Rules of Order Revised, 4th Ed., XII 70(a)
The second thing you will want to do (and maybe this is obvious) is find some like-minded individuals. At least ten would be a good starting point, depending on the size of your field (how many mycologists are there in Britain anyway?). If you already have a lot of connections in your field, take some time to meet with them individually and discuss with them why you think the British Mycological Society isn't working properly and what your vision would be for your own Society. Once you feel like you have a reasonable number of supporters, you should call a meeting with all of them together. They purpose of the meeting should be to come up with a shared vision of what the new society will be called and what its purpose should be.
You should also be discussing whether or not the Society will be feasible. Is there really a place for it next to the existing British Mycological Society? Will there be adequate interest from others in your field to make this a worthwhile endeavor?
Be careful about putting out some kind of public notice, lest you find yourself with the newly-formed society being taken over by strangers with different ideas from yourself.
Once you feel like everyone is on the same page and is actually committed to the idea of forming a society, you should move on to step two. If you can't get ten people excited about forming this society, then it probably won't last very long.
Third Step: Write your Charter/Bylaws/Constitution
Robert's Rules of Order Revised, 4th Ed., XII 67
Essentially you want to write a document that lays out the founding principles of your society, called a "Constitution" or a "Charter" or "Bylaws" (I prefer Bylaws myself). You'll see a lot of different preferences for what to call this document based on culture (here in Canada you will often hear the entire document being referred to as "the Bylaws"). At a minimum this should include:
- Name and object of the society. (What the name of your society and what is its most fundamental purpose?)
- Qualification of members. (Who is allowed to be a member of the Society? How can people become members? This would be a good place to include a non-discrimination clause. Are there member dues/membership fees? If not, where will you get your operating budget?)
- [Executive] Officers and their election. (Who will run the operations of the Society? What will the positions be called, i.e. President, VP, Secretary, Treasurer?)
- Meetings of the society. (How often are meetings held? When and where are they held? Who can call a meeting?)
- How to amend the constitution. (Who is allowed to change the document and under what circumstances?)
You may want to form a small committee (3 to 5 people) to draft the bylaws because it can be hard to coordinate 10 people or more all writing the same document. Once you have a document that the committee is happy with you should call another meeting and present the draft to your supporters. At this meeting your supporters should have the opportunity to discuss the bylaws and amend them if necessary. Finally you should put the acceptance of the bylaws to a vote. If the vote passes then congratulations, you've just formed a new Society! If you can't collectively agree on a set of Bylaws, then your Society will not last long.
Fourth Step: Elect your Officers
Robert's Rules of Order Revised, 4th Ed., X 58-62
Your society will need Executive Officers to handle the day-to-day organization and operation of the Society. The titles and roles of these offices should be laid out in your Bylaws. You will usually need a President, Treasurer and Secretary at a minimum. If no one is willing to volunteer for these positions, your Society will not last long.
Fifth Step: Promote and grow
At this point you have the skeleton of the Society, maybe some of the internal organs (are you the heart or the brain?). You need flesh out your society with more members if you want to make any kind of impact. Start advertising the fact that there is a new Society in town. Maybe get some posters on campuses across Britain. Try contacting the biology department secretaries of local universities and ask them if they could pass along invitations to the students and faculty. Make some profiles on social networking sites. Put together some small events, like a networking social in a public place (so you don't need to pay rental fees). As you add more members you can start taking on projects like branding.
Keep in mind this will be slow going. You likely will have to work for at least a decade, if not more, before your society really starts to gain real influence in your field. It will be tough to get new members, especially if you have membership dues, because people might be worried that it is some kind of scam. This is why you need a dedicated group of founding members who can stick with the Society for the long run.
Sixth Step (Optional): Get yourself officially recognized by a larger organization
This is not necessary but can be helpful. For student societies, you often want to get your society ratified underneath your student union at whatever university you belong to (become a member society). This is often helpful because the union will have resources to help protect and promote your society. By "protect" I mean that they will usually insure your events so that you aren't personally liable if someone is injured at an event, etc. as long as you abide by their bylaws. I am not familiar with any similar arrangements for non-student societies.