Edinburgh and thereabouts in June (advice sought)

The known parameters:

I have classes to finish and papers to write up until about 28 May.

I am attending the centenary of the World Missionary Conference (which is another blog post in its own right) in Edinburgh from 2 to 6 June.

I need to get to Hobart (maybe with a quick laundry stop in Melbourne) by 14 or 15 June.

A travel agent suggests either Qantas+BA to Edinburgh or Emirates to Glasgow.

This will be my first trip to the UK and when I visit new places I try to get out of the main destinations a bit, at the very least by taking a cross-country train trip. Edinburgh to London by train?

Apart from being a first-time tourist in the UK, I would like to catch up with extended family and friends in (or near) London. Should I spend a whole week in London? Or take some time out in a holy place after the conference? Ampleforth? Iona? Lindisfarne? I am geographically challenged – are any of these practical? Affordable?