Hi there, I will try and answer your questions:
Why do they want me to write an invoice as a Locum? Are they trying to run away from some responsibilities?
As a locum you will be responsible for your own taxes and you will not accrue holidays so maybe they are trying to get away from these liabilities. I would refute the argument you are a locum as you were specifically employed by them as a permanent employee and sue the statement to support that.
According to you, Dr, how much should I receive having in consideration £36.500 per year?
If you were on a salary of £36,500 a year then your weekly pre-tax pay is £701.92 and your daily rate is £140.38. You need to calculate how long you have worked there for to check what you are entitled to.
How about holidays and notice period?
You get 28 days holidays per year if you work full time. You accrue these proportionally over time, so if you worked there a month, you would have accrued 1/12th of 28 which is 2.3 days of holiday, payable at your daily rate above.
Your notice period is a week, again payable at the rate above.
Should I talk to them, Should I wait to see what are they going to pay?
I would wait and see what they pay but don’t leave it too long – they should not be delaying payment so if you have not been paid within a week, you should pursue them for the money owed.
And above all, anybody is free to lure people with promises and not keep their word with impunity?
Legally yes it is possible. As mentioned in the first 2 years of service you could be dismissed for more or less any reason. So this becomes a moral argument rather than a legal one.
Hope this helps?