All I can find are student visas.
The question of visitor volunteering is covered in Appendix V, Part 4A, which says (on 20 Feb 2016)...
A visitor may undertake incidental volunteering (i.e. the main purpose of the visit is not to volunteer), provided it lasts no more than 30 days in total and is for a charity that is registered with either the Charity Commission for England and Wales; the Charity Commission for Northern Ireland; or the Office of the Scottish Charity Regulator.
The key points are that the work is incidental and host has obtained a certificate from the Charity Commissioner. The latter is generally a show-stopper.
I looked briefly at HelpX and WorkAway and concluded (on assumption) that the hosts here were not bona fide UK charities. If they are, you should get their status confirmed in writing before you depart. If they ask you why you need confirmation in writing, you can print out this answer and email it to them.
The FAQ on HelpX gives the following disclaimer...
Many countries require a work visa in order to be a volunteer helper abroad. Several countries, including Australia, New Zealand, UK, Ireland and Canada have Working Holiday Schemes for young people from certain countries, which you may wish to investigate further.
The FAQ for Workaway gives this statement...
Workaway can not advise or organise travel or working visas. All volunteers must obtain the relevant visa (Working , Volunteering or otherwise) for the country they are travelling to. All questions regarding travel and visas should be put to the relevant embassy in the home country before travelling.
For volunteering activities that are longer than 30 days or with a non-charity, you need a Tier 2 visa or a Tier 5 Government Authorised Exchange visa. The point is that the UK considers those activities to be work and not volunteering in a legal sense. T2 and T5 GAE are dealt with on the StackExchange Expats site, we handle only volunteering as defined in Appendix V.
It pays to play it by the book when visiting the UK. Last week we had a removal from port and 10 year ban where the person had done some babysitting previously. The removal was on gravely serious grounds and babysitting was just icing on the cake, but you can bet that the babysitting hosts got a warning from the police along with a visit from child welfare.
As noted your follow-up questions about T2 and T5 can be posted on Expats.
Note: this answer includes Americans, Japanese, Brazilians etc along with visa-nationals (Egyptians, Indians, etc). It works the same way for all nationalities.
Note: EU nationals can do what they want as long as they are exercising treaty rights.
Note: the UK's Youth Mobility Scheme (the current branding for 'working holiday maker') can also be used to perform volunteering lawfully. You wrote that you're American and thus not qualified to participate in the YMS.
Note: for Americans the relevant 'embassy' for enquiries is the British Consulate General in New York, or call the relevant UKVI contact centre (or chat facility).
Note: Canadians are part of the Commonwealth and thus eligible for YMS.