It is puzzling that they both pleaded guilty. They also both represented themselves in court, so it seems they weren't getting any legal advice at all.
Perhaps they weren't entitled to legal aid? I was under the impression that anyone charged with a criminal offence was entitled but perhaps I'm wrong.
Maybe they just wanted to get the whole thing over with quickly and not face further embarrassment? It's a good example of how these types of charges could easily be used by the authorities to intimidate people. The damage is done whether they are convicted or not.
To answer your question, it seems you can appeal regardless of how you pleaded [0]. But you have to do it within 28 days.
[0] https://www.gov.uk/appeal-against-sentence-conviction/crown-...
