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.
Have they increased the monetary value of their offer now they are asking you to be a contractor? Being self employed means that you lose the benefits of having an employer: you will need to pay your own employer's national insurance and you won't get holiday or sick pay, etc. etc.

As a rule of thumb, as a contractor you should be looking for roughly double the salary that you would receive as a full time employee.