England & Wales unpaid invoice: can I make a court claim for money?

Yes. GOV.UK says a person or business can apply to the county court to claim money they are owed. The claim can often be started online or by post, and mediation may be available. Before court, it is sensible to send a clear demand and preserve the contract, invoice and correspondence. Official source: GOV.UK Make a court claim for money. Guide: https://getinstantlaw.com/small-business/customer-wont-pay-invoice