I agree entirely that Heathrow's main line rail provision is poor. Once HS2 and Crossrail are fully open that will address access from the north, albeit with a change needed at Old Oak Common which is a bit of a hassle with luggage though fine if HBO. Had the proposed spur gone ahead, this would have been resolved but my understanding is that the perceived demand to be too low to justify the loss of capacity into London directly (and a Heathrow loop, rather than spur, would have cost a lot more as well as made end to end journey times less competitive). My own preference would have seen an HS2 spur continue to Gatwick and Ashford/Brighton, because that would address not just inter-airport connectivity, but connect areas south of London to Heathrow, the midlands and the north of England - something which is a right pain in the neck by train at present. Ah well...

As for Edinburgh, there is now the Edinburgh Gateway station close to the airport which has a tram link. I had hoped they would build just south of the end of runway 24, roughly between Turnhouse Rd and Pennymuir, where there are derelict (RAF?) buildings/hard standing. However, I don't believe the intercity trains stop there, which would mean that your friend would have to trundle through Fife on a stopper. From memory there's an extortionate add on for the tram fare to go the final bit to the airport, too.

Britain and meaningful public transport to major airports is sadly lacking. Compare it to Amsterdam, Zurich, Geneva, or Frankfurt, and we're decades behind.