The easiest way to actually provide your services to a charity is volunteering for the charity themselves; it's free, and off-the-books. You don't have to declare anything, and you don't have to pay national insurance, which is a plus.
However, if you want to incorporate as a nonprofit in the UK, you need to incorporate as a Company Limited by Guarantee (CLG), and then file articles of association (more here).
Nonprofits in the United Kingdom have to declare their NPO status in their articles of association (you will have to write them from scratch or have a lawyer draft these), and have a set of standards which the organisation will adhere to (how their profits will be spent), which are reviewed by Companies House. There is no specific (i.e. it has no specific document) way to register a non-profit in the UK that isn't a charity. (Another alternative is a Community Interest Company, but I don't know if it applies here)
As for tax itself, you will still have to pay tax as a company, but your taxes are relieved on the following:
- Charitable income
- Rental income
Provided these are on the list in your articles of association (source).
In my opinion, unless you can make a charity, it's likely better than you simply volunteer, or incorporate as a Private Company Limited by Guarantee and list yourself as the sole guarantor, if you really must incorporate.
You will, however, be able to take out loans and wont be liable for any more than what you put into the business at incorporation. You might want to Check out OnStartups Stack Exchange if you fancy making a company, though.
http://www.charity-commission.gov.uk/ is the site for registering a charity in England or Wales.
I currently operate as a sole-trader with books showing zero profit. Is this a problem? (It's not faked!)
As a sole trader, you keep all profits anyway, and you aren't audited, so this isn't a problem.
Disclaimer: I'm not a lawyer, but I did go to business school in the UK.