all admiral itineraries assume that whatever luggage you have is maneuverable at speed through crowds by the personnel (travelers) deployed on your mission (journey) - in other words, once you pull luggage off a carousel - it's mobile and does not require additional humans, carts, or doubling back - you and your party are mobile and capable of schlepping your own shit - no children or elderly complications - those bring travel command and control to it's knees; in addition, this is not a penelope junket, but YOLO, and no cost is spared.
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1. fly in to edinburgh: NO
never switch domestic for transoceanic, for a dozen reasons
scotland does not currently and has never had nonstop service from dfw or iah
day 1
you are in austin - don't lengthen your trip by 8 hours minimum and risk a domestic irrop trying to switch in atlanta, newark, jfk or boston - use the best option - nonstop to lhr
day 2
arrive T5 with rooms booked at the attached sofitel hotel with prepaid breakfast included the following morning; take a hot shower, stay awake as long as possible, and reset your circadian; when the bar opens in the hotel, time for pints and early pub dinner. it's not worth the hour tube in to london and hour back - you are hanging out at heathrow today - we can discuss options offline for killing hours noon-4pm
day 3
get up the next morning early - force yourself to get up at 530am and walk in when the incredible sofitel breakfast opens at 6am
don't fly to scotland - splurge on a car service to kings cross, tip everyone to handle your bags - and book first class on the LNER Azuma train to Edinburgh Waverly departing every 15 minutes it's an intercity train introduced in 2019 that takes 4:25 at 125mph; the first class lounge at kings cross is quiet and not over-run by mid-morning, and then you get free lunch on the train
you booked 2 nights at the balmoral hotel in edinburgh which is connected to and sits literally on top of edinburgh waverly station - let balmoral know you are arriving by train and they will send a porter down to meet you and collect your bags
book dinner the first night in the hotel at number one: https://www.roccofortehotels.com/hotels-and-resorts/the-balmoral-hotel/dining/number-one/
day 4
because you are relaxed, rested and ready, and you are right in the middle of it all, you can do edinburgh in 1 long day; START EARLY, which in the summer in scotland is easy, because it only gets dark for 3 hours
day 5
because you are staying at the train station, you are pre-positioned to take any of the royal scotsman trains, which all depart from and return to waverly:
https://www.belmond.com/trains/europe/scotland/belmond-royal-scotsman/about
i'll pause here - pick your next train and then we'll pick it up from there