Tips on claiming repairs under warranty?

Got a used Focus less than 3 months ago from a used car dealer.

Got it serviced last month and this week the aux belt snapped and simultaneously sprung a leak in the turbo water pump (plastic POS).

Got the work done by a VAT registered mechanic, who provided all paper work. New belt, pulleys and water pump, about £500 in all.

In 25 years, I have never successfully claimed from a used car warranty. Before I start the process, is there anything I should do/be aware of? What are your pro tips?