While I don't think physical sales in brick-and-mortar stores are going anywhere, I do thing there will be a significant fallout between most specialty video game stores and their customers at some point. First, let me point out that I don't know the ins-and-outs of GAME, because I live in the US; I've only dealt with them when I imported Xenoblade (and it was a smooth experience on my end).
First off, if GAME is anything like GameStop, then chances are in-store experiences aren't great (if borderline terrible in some locations). GameStop consistently has its employees push the Game Informer Magazine on customers, along with preorders and buying used. While in many cases it's not bad, and in few the employees are actually smooth and nice about it, some are outright abusive (I've been personally insulted before because I didn't buy a used game and for buying a Pokémon game on seperate occasions). Couple that along with employees trying to pester parents into buying something their kid(s) didn't want. The in-store experience is key, and if it's anything like GameStop, poor experience means customer loss over time which means to sales loss which leads to poor credit and publisher relations. Not to mention purchasing online completely does away with any of these potential issues, and is more convenient in most situations.
You can also match this with the fact brick-and-mortar stores usually have an agenda in regards to their stock and sales. You will most likely see loads of stock of Call of Duty XXX, but rarely will you see a niche game published by XSeed or Aksys. The games published by XSeed and Aksys are largely available online, and sometimes even have lower prices. Take Shin Megami Tensei: Digital Devil Saga. It was near impossible to find in most stores at release here, but you could find it everywhere online. Now stores like GameStop sell it for much more than it's actually worth. You can also add poor pricing of common games, even though browsing the GAME online store, it didn't seem that was the case at all (I imported Xenoblade for less than it would have to buy domestically!).
Finally, you can also attribute the marketing machine to this. While I can't really see any ads or marketing for GAME without going out of my way to see them, my thought is their advertising and marketing machine is likely very poor and broken. If so, it'll lead to very small numbers of new customers. Poor promotions and sales that make little sense and save customers little money will do little to attract them. For example, my local GameStop ran a promotion for 5% off select games. The 5% off was only on sports games, and used ones at that, which are games no one wants (used copies of Madden and such fetch prices of about 25¢ - $20, depending on the year). Ads that have no defined target or ads that target a group but fail to deliver the message are largely ineffective where they need to be. They probably also don't take into account the other effects the advertisements have on potential customers.
To be honest, if GAME has any combination of the above issues, it's not hard to see why they would have a fallout with their existing customers and fail to draw in new customers. Not to mention the relative ease and convenience of ordering online as well as excellent sales (Amazon daily deals can be absolutely amazing) make ordering online much more attractive.
However, like I said, I don't think brick-and-mortar stores are going anywhere anytime soon. For one, people are impatient for the most part and want their stuff now. Ordering online means you have to wait a few days to up to two weeks to get your game. Second of all, if the UK is even as bad as it is here, when ordering a GBA game online can be very risky as you run the chance of getting a counterfeit copy (niche point, I realize, but it can still happen, even for non-GBA games). Third of all, at least here in the US, shipping an item runs the risk of it being lost or damaged in transit. If anything, I could see GameStop or GAME being replaced if either of them goes under.
