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I am about to go on a quick jaunt to England where I will be flying to Heathrow and from there taking a train to Swindon. I pre-ordered my tickets online, paying through PayPal, and was told I could pick up my tickets at a ticketing machine as long as I had my reservation number and a valid payment card.

I am arriving from the Netherlands, so passport checks should be minimal.

How does this actually work, and are there any common pitfalls I should be aware of?

I can't answer 2, but 1 I can. This page tells you where to find the machines at each station. Note that ticket offices CANNOT usually print pre-purchased tickets; you must use the self-service machine. You find a machine, select the option to pick up pre-paid tickets (I don't recall which software the Heathrow machines use so I can't instruct you exactly). It will probably prompt you for a card at some stage; since you paid with PayPal you can insert any chip-and-PIN card. You shouldn't need to enter an actual PIN. Use the on-screen keyboard to enter your Tickets on Departure code (referred to as a booking reference; an alphanumeric code usually in the subject line or else displayed prominently in the confirmation email, depending on who you bought the tickets from). Your tickets will then print; sometimes if you have multiple coupons (eg a return ticket with seat reservations in each direction for the Paddington-Swindon leg would have four coupons, possibly plus a collection receipt) there can be a notable pause between printing coupons; please ensure the machine is completely finished before you walk away! Then it's a simple matter of getting on the train and showing the guard your ticket.

I assume you are booked on the Heathrow Express (or Heathrow Connect) to London Paddington and changing there for a Swindon train. At Paddington some platforms have automatic ticket barriers, so depending on which platform your Swindon train is leaving, you may have to use them. Insert the Heathrow-Swindon ticket (note: not your seat reservation coupon, if you have one) into the slot on the front of the barrier with the printed side face-up (and the magnetic stripe face-down), and remove your ticket when it comes out the top (you need to keep your ticket until the end of your full journey). If the barrier fails to open after removing the ticket, find the member of staff manning the barriers (they will usually station themselves at the wide luggage gate) and show them your ticket; they'll let you through manually.

If your ticket type is "Advance" (this will be printed prominently on the ticket) it means any reservations you have are mandatory, and you must use the train booked in the reservation, unless a delayed train causes you to miss your booked connection (in which case you will be allowed on the next train). For legs of your journey without specific trains specified on the tickets and reservations, you may use any suitable train. If, on the other hand, your ticket type is "Super Off-peak (Day) Single/Return", "Off-peak (Day) Single/Return", "Anytime (Day) Single/Return", you can use any train, provided you comply with peak-time restrictions if your ticket is off-peak or super off-peak (newer ticket styles have a web address you can visit to discover what these restrictions are; older styles have a two-character code printed near the word "Validity", in which case you can go to http://nationalrail.co.uk/XX where XX is the two-character code). If you tell me the exact journey you're making and how much you paid I can probably find this out for you now.

Going to Swindon, the trains currently in use are from the 1970s and have manual doors which can take a bit of getting used to for tourists. They're soon to be replaced with newer ones, so depending on when you're travelling you might not have this problem. If you find yourself at the front of the queue to open the door on the train, you need to lower the window, and use the door handle on the outside. Wait until the guard has unlocked the doors first (a "Doors unlocked" light will switch on inside, and a large orange light on the outside of the train). Similarly, if you're the last person getting on or off the train, slam the door shut behind you and save the dispatch staff a trip down the platform.