I think seeking legal advice from the right source would be the key. Simply because there is a legal department does not mean that there is expertise there to deal with the issues raised. There are, from my reading of the situation, two issues: one of them is about citizenship and the other, which may well flow from this, is the question of the legality of the adoption.
With regards to citizenship the issue is much simpler. Since 1983 children born in the UK or in the airpace of the UK cannot automatically become UK citizens or subjects. In addition to this there have been a raft of changes to the immigration legislation and guidelines. If I am not mistaken, a child would have a right to UK citizenship if he/she was born after 2006 and that one or both of the parents was a UK citizen, irrespective of the status of the relationship, i.e. the issue of half or full blood has now forever been resolved. In lay man's terms, a love child now has the same rights as one born within wedlock.
With regards to which court has jurisdiction in this case, it is very unlikely that an international court would bother to get itself involved. Remember the Human Rights Act 1998? Well, this was enacted to 'give effect' to international human human rights when Tony Blair said he was 'bringing Rights Home'. This makes our British courts a public authority. Accordingly they must interpret the law in ways, as much as possible, that reflect human rights. Failure to do this would give rise to a challenge, which may then be taken up by the European Court of Human Rights.
The more difficult issue here for me would be defining June's position. Would she be acting as an advocate or would she be acting as a local authority official? In my experience, most local authoties no longer want their employees to act as advocates, as this apparently causes conflict of interest (this is quite strage because during and after my training as a social worker the main employers of social workers have always remained local authorities. Despite this I was taught that one of the roles of a social worker was to advocate on his/her client's behalf).
Because of the apparent conflict of interest it is unlikely to get any reasonable advice from the local authority's own solicitors. Only an independent solicitor would do. I would therefore start with the Immigration Advisory Services (IAS), who may point June in the right direction. Failing this, the parent in question may also have his/her own solicitors.