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: 
Tell them that **this is not what you signed up for**. They have gone back on your agreement and violated your contract, before you've even started work. Say **you regretfully must decline to work with them**.

And when they come back at you with nonsense about how you've signed a contract, remind them what that contract says. No court is going to side with them defending a contract that you're walking away from *because they have completely broken it already*.

Any further discussion takes place through your lawyer, *only*.

In the meantime, **call up your old supervisor, tell him it didn't work out, and ask whether there is room for you to return**. It hasn't been very long, so that may very well be possible. It might sound strange but **there's really no shame** in simply returning to your job. I work with three people who have gone off and varyingly worked for other companies for a bit (later realising their mistake) since they first became my colleagues.
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.