If you want to get to Edinburgh from LHR, then it would be a lot less hassle to fly. I can see loads of availability on BA for 5 September (and I’m not logged in – no status), but prices are higher than I would expect. (Is there some big sports event at Murrayfield that day???)

If you do stick with the train. Getting to Kings Cross is straightforward – direct on Piccadilly Line. A single ticket on the tube (cash fare) is £6.00. If you use “pay as you go” on either Oyster or a contactless bank card, it’s only £3.10. So – if you’re going to need an Oyster for the London stay, you might as well use that from LHR and save.

For the stay in London. A one week zones 1/2 travelcard is £32.10. If you use PAYG (on Oyster or contactless bank card) you pay a maximum of £6.40 per day. If you know you’re doing a lot of travel every day, then the weekly season ticket saves money, but if you’re likely to have some low travel days, then PAYG may be cheaper.

The key message is don’t buy cash fares. A single for the tube is £6. A one day travelcard issued in paper form is £12. And to reinforce the point about avoiding cash fares – it isn’t actually possible to pay cash on the buses.

Using Oyster is generally the simplest thing. You can load a weekly season and/or “pay as you go value”. The only exception is if you want to use the “two four one” discounts. That is only possible with a national rail ticket – so to get that benefit you’d buy a weekly season on paper from a national rail station. It requires a passport size photo but costs the same - £32.10

For UK citizens, using a contactless bank card is even better than Oyster. No need to obtain a separate Oyster card yet same prices as Oyster for PAYG usage. If a weekly travelcard would be cheaper – the system will “cap” the charges to the equivalent of a weekly travelcard. However, overseas bank cards are not guaranteed to work (some do – some don’t) and you may be hit with foreign currency transaction fees by your bank.

Rail fare from London to Edinburgh.
A “walk up” fare (flexible – any offpeak train, which includes the entire weekend) is £125.70 single or £134.20 return.
The cheaper tickets are called “Advance” (it’s a specific ticket type) and can be as low as £20 single. There is no guarantee that the very cheapest fare will be available on any particular date – and indeed as September is still quite busy I wouldn’t expect any of the very cheapest fares to be offered. Other fares will be offered, as the cheapest “fare buckets” sell out, more expensive ones will be available. These tickets are non-flexible – zero zero zero flexibility – you can travel on the service booked. Travel on any other train and you’ll be charged a walk up fare.
The railway industry generally doesn’t release “advance” fares until 10/12 weeks before travel date.