There are a few issues here.
SDO, CSDE: Firstly, some platforms are too short to fit all of the doors on, especially with the lengths of trains being extended. Thus at some platforms it is necessary to ensure that the first or last doors of the train, or both, are not unlocked.
The London Underground has its own system to do this (https://en.wikipedia.org/wiki/Selective_Door_Operation ). Here it appears they decided to use GPS, but the LU almost certainly uses a much simpler (and more reliable system).
Of course, if a GPS fix isn't obtained, you don't know whether you're at a short platform or not, so you can't assume anything.
A related safety issue is whether drivers open the doors on the right side of the train (!). Yes, believe it or not, they sometimes get this wrong. When the London Underground moved to one-man train operation (meaning that the doors were controlled by the driver, rather than by a separate guard), they had some issues with this happening. They responded by introducing a simple, low-level system called "Correct Side Door Enable", which uses an electrical loop by the side of the track in platforms, which is detected by the train and allows the train to determine that a) it is in a station and b) which side the platform is on. This prevents the driver from opening the doors in error.
You can find some information on CSDE here: http://www.trainweb.org/districtdave/html/correct_side_door_...
I would guess that the LU's SDO system uses some similar or even interrelated mechanism.
Metro vs. Rail: Note that on metro trains (e.g. the majority of London Underground trains) the doors usually open automatically, whereas on other trains you have to press a button to open the doors once it illuminates. This makes the potential consequences of accidentally opening the doors on the wrong side worse on tube trains, especially when you consider how packed they can get. Whereas with a train like the class 377, a driver would have to unlock the door on the wrong side and then the passenger would have to accidentally open the door on the wrong side. So the consequences of accidentally unlocking the doors on the wrong side is not -that- severe, which makes the apparent difficulty and obscurity of the overrides available to the driver (plus the implication that the drivers aren't properly trained on their operation) all the more ridiculous.
(A particularly curious variant is found on some trains of the Paris metro; the doors try and open as soon as the driver tells them to, but a mechanical latch on the doors prevents it. A passenger has to move up the latch before the doors can open. The latches were probably retrofitted, perhaps to mitigate unnecessary heat loss, but it also serves as a safety measure. This system has the advantage over a button-based system that a malfunctioning microcontroller cannot randomly open a door at inopportune times, though this is a rather academic advantage given that I am unaware of such cases.)
I have myself ridden the 377, and on one occasion I did notice an unusual delay before the doors unlocked, of about a minute, which involved the "door out of order" lamp momentarily flashing on and off a few times, perhaps suggesting some sort of train reboot.
I find the term 'GPS repeater' strange. The nature of GPS would lead me to believe that 'repeating' GPS would be rather difficult, though I could be mistaken. It seems more likely that by 'GPS repeater' what they actually mean is some sort of overriding 'you are at station #27' signal, perhaps implemented much like the LU's CSDE system. The Wikipedia page for SDO seems to confirm this (https://en.wikipedia.org/wiki/Selective_door_operation - note that 'Thameslink' and 'First Capital Connect' are effectively synonymous).
The idea that not being able to quickly open the doors is a safety issue seems quite peculiar to me. On the 377 and all non-metro trains, all doors are fitted with passenger-operable emergency releases. There are also somewhat anonymous emergency releases which could be operated from outside the train by platform staff. Both such releases are mandated and governed by railway standard GM/RT2473 (which can be found here, along with some rail accident reports referencing it: https://www.google.co.uk/search?q=gm/rt2473 )
That said, there is one way in which the Class 377's doors are implemented in an obviously deficient way: if you hold down the open button before the doors are unlocked, the doors will not open when they are. You then have to release and re-depress the button. In other words, whoever programmed the door erroneously chose an edge-triggered behaviour rather than a level-triggered one. This is in contrast to the buttons on the Class 319 (which the Class 377 is replacing), which behave correctly. I previously ranted about this here: http://www.devever.net/~hl/train377