Unfortunately, the Typhoon costs a lot more money and doesn't have the VSTOL capability needed on the current British carriers.
I have a feeling in the end this argument is mostly about sub-contractor work and long term maintenance contracts, aka jobs for UK citizens. There will be a compromise solution and the UK will be flying Lightning.
To be fair, their new carriers will be able to operate catapults and conventional aircraft should they need to.
They are not currently being built with them, but the design is supposedly able to to accomodate them.
They'll be part of an EU navy before their careers are done, hence why giving them the ability to operate American aircraft totally independantly is not a great idea...