There is a cultural gap here, because contemporary London is a time and place where we assume that:
- There is a process,
- which is documented,
- which anyone can follow,
- and which ends up with everything working.
We are in a low-context society where most day-to-day government processes are accessible in this way. We can submit a form without ever knowing the name of the official who will process it. We assume, as background, that there is a level of basic competence in public service that will enable these requests to be serviced fairly (and there would be some sort of other equally established procedure to follow in case they were not).
Victorian England was in many respects not like this. To get things done, you don't just fill out a form. You are in a high-context society where you have to know people and be someone. If you are setting up a foundation to look after the stray dogs of Lewisham, then you'd better be on good terms with the local worthies. You are navigating a complicated system where the formal rules are only part of the story. Charity was an expression of social character and advancement, presumptively operated by the wealthy and virtuous. In the mid-century it was becoming more "professionalised", especially in the capital, but it was still basically rooted in the idea of notable people of good character dispensing their largesse on (a worthy subset of) the needy.
Accordingly, I imagine that a 19th century Londoner would not make statute law their first port of call, although they could research it themselves from first principles, if they were very persistent. If they did, their main problem would be that charity law was largely uncodified, and dependent on a body of case law. It was a complex topic and the subject of a good amount of political attention. People might become aware of the debates over charity management and regulation through general gossip, or by reading newspapers: this would give them an understanding that the formalities of setting up their foundation "properly" would be important.
The putative unassisted founder would have to understand trusts quite well, which are not defined in statute, but are reverse-engineered from historical practice - not necessarily resulting in a body of law which was coherent. This is also before the fusion of law and equity, so there are several parallel possibly-inconsistent sets of rules in play. It would be difficult to come to grips with the body of raw material. For example, the Charitable Trusts Act 1853, which established the Charity Commissioners in an effort to bring some oversight to the situation, was inapplicable to a large number of charities affiliated to the Church of England, or to those run by subscription as opposed to from an endowment. (Most new charities at the time were subscription-based.) A layperson reading the Act might come to all sorts of wrong conclusions, while still not being equipped to understand what a trustee can and can't do.
Plausibly, anyone in a position to contemplate setting up a new charity would know someone who was already involved in some sort of charitable endeavour. London was full of them, even at the very local level of vestries (parish councils) through which the metropolis was governed day-to-day. These boards were simultaneously responsible for church affairs, administration of the Poor Laws, and various local-government functions; and might have associated trusts enabling them to hold an endowment and distribute funds. The people most likely to be on those boards would be those citizens who were also most likely to either run charities themselves, or at least be in contact with them. They might not be legal experts as far as the nuts and bolts of incorporation and management, but they would have a working knowledge of how to go about things. Perhaps your characters will first consult a vestryman they know, who could give specific advice or direct them onwards to a friendly expert. Similarly, if your characters have contact with a specific charity, such as a lying-in society or a school, then they could ask the people who run it.
If the proposed foundation is something bigger or more specialised, such as a hospital, then the necessary people would be less accessible. But it would be reasonable for someone wanting to set up their own hospital to seek out someone who already had one, perhaps through a chain of social connections (I will ask Rev. Green to introduce me to Dr. Watkins, who knows Sir Anthony, ...). The process of consultation is not just you asking for advice, but proving yourself to those people as someone worthy of support. It's not a one-off event but the formation and maintenance of social relations.
For actually creating the organisation in a formal way, one would probably consult a solicitor if there was any concern about doing things properly. Again, though, one might find the solicitor because he's Rev. Green's old university chum, as opposed to choosing one at random. It would also be possible to proceed informally for a time, but the more money was handled, the more disputes between the people involved, and the more the attention of the state was engaged, the more problems might arise. Many charities operated for years before incorporation, for example originating as a committee drawn from a church congregation and subsequently spinning off independently.
Some further reading:
- Chapter 13, "The Deformation of the Gift", in Outcast London by Gareth Stedman Jones (OUP, 1971). This is about the Victorian-era conflict between a traditional view of charity as the personal benevolence of the local squire, and the emerging view of impersonal benevolence where a charity acts as intermediary, disconnecting the donor and recipient. We can see in-built suspicion of philanthropic efforts that aren't rooted in the "proper" existing social relations.
- Patterns of Philanthropy by Martin Gorsky (RHS, 1999). Same era, different city (Bristol), but covers a lot of modes of social organisation.
- Women and Philanthropy in Nineteenth-Century England by Frank Prochaska (OUP, 1980). Invaluable for showing the attitudes and modes of operation of the women who were doing a lot of the work; accessibility of philanthropy and volunteerism to the middle classes; and a spectrum of respectability to radicalism of the causes supported.