As well as the potential for horrible physical and economic exploitation and depressing sexualisation of this issue by adult men, selling human milk doesn’t appear to be falling under routine health risk monitoring protection nor flagged up under consumer protections around food safety. 
It doesn’t look like breast milk sale was included in the anti-exploitation legislation banning the sale of other human body products (like blood or organs sales are banned in the UK even where these are matched to recipient and health-screened). These laws prevent exploitation of human bodies and prevent risks to the provider of those body products who is selling them to someone else. They also prevent unsafe or contaminated supply to the end consumer because of the desperate and therefore highly unhealthy situations of people who may be driven to sell their own body products.
Is breast milk and women and babies with an interest in this left unprotected, then because women’s bodies are commodities to exploit, so that’s fine? Or because lawmakers assumed that only a few women would have gone in for this; so it’s not worth spending Parliament’s legislative time on it? Because lawmakers don’t want to think about poor or desperate women and babies and what they might need to do to get by financially? Because as OP says there doesn’t seem to be much research around this and so perhaps lawmakers wouldn’t know or guess about the health risks of selling human milk as food to babies or adults? Because they also assume that all mothers are completely interchangeable with each other so all milk is the same- like we assume with agricultural dairy products?
Always worth thinking about who and what gets legal protections and why those laws might have been put in or not. Due to naivety and sexism/misogyny it looks like there can’t have been any law making in the last few hundred years around selling breast milk commercially. Choicey-choosey people will try to hark back to the good old freedom days of destitute women offering themselves up as commercial wet nursing to richer families, but really, is that what we want to be legimitising for financially unsupported women in this day and age? Where is the political focus on society supporting new mothers so they aren’t desperate for money?
Food safety sales laws at least already give some legal powers to stop the commercialisation of milk as an adult human female body product. Current laws require sale of human body products as food must be to specified legal standards; presumably to avoid passing on infections to the recipient (and you could argue in this specific case- new laws should prevent sale because production could cause harm to the producer and her family).
I just found this page by googling so I can’t vouch for them but maybe it’s worth reporting this to Baby Milk Action who say they monitor baby food (including breast milk substitutes) companies’ actions and set out the legal rules for companies around production and fair advertising. www.bflg-uk.org/uk-laws
www.theguardian.com/sustainable-business/2015/mar/02/breast-milk-market-uk-bodybuilding
This Guardian article says that it’s not illegal to sell human milk in the UK but that sellers must be able to demonstrate specific legal standards in this food production. Which it would be extremely difficult for any of them to do.
So maybe an effective action to take against commercial selling could be write to the Food Standards Agency, copying in your MP and asking them to lobby the government to bring in regulation around this specific issue and informing them of the companies or websites already selling this as a product? That way the Food Standards Agency can investigate and use their existing standards to get companies to demonstrate how they are meeting these standards (and from the looks of the FSA quote in the Guardian, companies won’t be able to meet that standard).
Also important to ask MPs to bring forward specific new legislation to protect women and babies from exploitation and think carefully about the effects of society changing the view of human breast milk into being an exploitable human body product in this day and age.
We’re not allowed to sell our own blood or organs or bone marrow in the UK, but we can donate them to those in need, which is brilliant. Breast milk donation via hospitals etc to babies who need it, is also brilliant. But why is is breast milk allowed to be openly sold? Is it because only women make milk, but men make blood and organs too?
Enforcement and new legislation might not stop underground selling online but could help to stop companies advertising sale of breastmilk in the UK and thereby making that sale look legitimate (ethically, and in terms of safety).
Food Standards Agency reporting link here:
www.food.gov.uk/contact/consumers/report-problem
Find your MP here:
members.parliament.uk/FindYourMP
Obviously there is also the possibility of male bodies with the aid of powerful medications lactating and selling on that milk as a food product. So logically the same legal restrictions should be applied to men selling on their milk just as they would to women. Even though this is far greater an issue centred around female economic deprivation and exploitation, and the absolute focus should be on not leaving mothers so desperate for enough money that they feel they need to sell breastmilk.