Both the vaccination status and covid test can be shown on your phone, but you must show the full certificate - e.g. download the pdf certificate from the NHS app or the pdf certificate from the test. Depending on where you board you may need to show them multiple times (e.g. at the baggage drop, inside the terminal at a health check desk and then finally at the actual check-in).
"The original test results certificate and vaccination certificate must be presented at the terminal (paper or electronic format) and must be in one of the following languages: English, Italian, German, French or Spanish. The COVID-19 test certificate must contain: guest’s personal data (verifiable with the other travel documents), date of the test, identification / contact details of the centre that performed the analysis, technique used and negative test result."
"Test results on an SMS or email will not be accepted"