Question: I was offered a job by a US software company.
On my first day at work they told me they couldn't actually hire me for legal and technical reasons.

So, now they want me to be a consultant and to become self-employed.

I had plans to buy a house in UK soon, and now I won't be able to do it unless I provide at least 24 months of invoices as self-employed.

Nobody is taking responsibility, they are blaming their 'solicitors'.

Therefore, my question is: how can I handle this situation (sudden request to change contract) in a professional way?

Answer: 
As you've already pushed back on the T&Cs for the consultant role, it sounds like what you've got is pretty much their final offer, so it's now up to you to decide what you want to do. You've got three options:

* Take the consultant role, with the intention of staying in the role long-term.
* Take the consultant role, but start looking for a new role ASAP.
* Reject the consultant role.

Any of these can be handled in a professional manner: the first is easy, the last is simply "Sorry, I'm looking for a role as a permanent employee". The second is perhaps the trickiest, but so long as you do your job as well as you can for the period you're working for your employer, it's going to be "just one of those things". Your potential employer has screwed this one up *massively*, so you shouldn't be expected to go out of your way to deal with their HR mess, and you've got to earn money in the meantime.
* I would find it hard to work for that company after these events. So forcing yourself into it should probably not be what you want.
* The issue is pretty clear; they stepped out of the contract on day 1. What does your contract say about how that works? Does it have a clause where they have to keep you on for at least XXX amount of time?
* Signed contracts cannot simple be "changed" by one party. Get a lawyer, get at least your costs back from them (that means the missed income during the time you are without job, etc.), and make perfectly clear that you will not be their victim. Depending on whether U.S. or U.K. law applies (both of which I'm not terribly familiar with) you might have more or less leverage.