Sorry, I'm not 100% clear on this - you worked for a company for 3 weeks and then they let you go, this was a week ago, is that right?
When should you have been paid if you were still employed by them? At the end of March?
If you've now put in an invoice many companies will only pay this 30 days after the end of the month that it's submitted. Ideally you should have put a date on the invoice that you wanted to be paid by.
You did actually have a contract with them, albeit an oral one plus whatever was written in any letter confirming your appointment.
Did you also claim in your invoice for your notice period, probably a week?
Personally I'd start ringing them up and asking them to sort it out asap.