I had a part-time co-founder but he is leaving. How can he hand back equity?
I'm based in the UK and I've heard of a form called J10, but once this is done how do we change the information registered on CompaniesHouse?
The company can always buy (back) equity from, e.g., a departing co-founder. So long as both sides agree to the transaction, no problem. This is just a straight sale of shares at an agreed price (often, but not always, at the price originally paid for the shares). The shares are held 'in treasury': the company owns the shares, unless it cancels them.
This handles the legal side of the transaction. As for the accounting form that you fill out to report changes in equity to the government (which presumably you do only for transactions over a certain percentage in the company stock), I can't help with that.