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: 
You can start your one-man limited liability company in the UK and use it to maximise your income by being tax-efficient. 

Instead of a monthly salary, you would agree on a daily rate. I'd say that £100 gross daily rate (that is you pay all your cost, taxes, etc out of that rate) is equivalent to about £15,000 annual pre-tax income (your official income where the employee withholds tax and employee's national insurance). 

Any threats of suing you are void, as long as you make sure any contract is between them and your limited liability company, and that there is no money in the company by paying out everything as dividends. Which is not optimal for tax purposes, but what can you do. 

BTW. No need to handle this professionally. You need to handle this in the way that is most beneficial to you.
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.