First of all, if your partner is self employed he must know that he's unlikely to be paid immediately a job is done. Most companies pay 30 days after the end of the month of the date of invoice. Some may pay quicker, others slower.
Did he agree payment terms before taking on the work?
3 weeks is quite soon to be thinking of going to Court quite honestly. You can start hassling for the money now but it would normally be 30 to 60 days until you get paid. Have you written to the company, sent statements etc?
You can make a claim via the small claims Court here
https://www.moneyclaim.gov.uk/csmco2/index.jsp