Hi Saj,
TfL services are not covered by the normal National Rail delay repay so in this case your journey is treated as being in two halves. If the Thameslink journey was delayed by 15 minutes then you might have a claim with them, but they will look at your arrival time at Blackfriars/Farringdon and compare the time you arrived at City Thameslink with what you should have expected at that time. The Underground and DLR also make refunds if you’re delayed by 15 minutes or more. If they were the main cause of the delay then use the Oyster online site and select either Blackfriars or Farringdon as the destination.
Good luck.