Well 308 won't 'die' anytime soon, and it'll likely remain the cartridge for the TR / MR / Palma / FTR disciplines as long as they are in existence ... with one possible proviso only. That is if NATO adopts a different set of cartridges in a different calibre, that may cause a rethink just as the move from the 19th century cartridges to 7.62 in the late 1950s did. Even there with the weakening of target shooting to military connections outside of the USA anyway, that might not be enough to cause changes.
I've been a competitive shooter for over 50 years and the changes over that time are enormous, primarily in an explosion in the number of disciplines. Outside of Bisley and a relatively small number of large competitions, 'Target Rifle' is dying. Regional clubs that produced 20 plus competitors for in-house matches back in the 1980s are lucky to get a half dozen these days, and regional 'Open' comps have nearly died. The YRA's twice annual Spring and Autumn open TR meetings at MoD Strensall use four of five lanes these days, and one of those is for F-Class. In the 80s, even 90s, two adjacent ranges had to be booked to accommodate a TR-only field.
PSSA at Diggle which used to have a large TR shooting complement, has maybe 10 now of whom half a dozen are regulars and that's despite TR shooting alongside F-Class on a two-weekly basis for a good local championship. The many northern inter-club TR or Palma challenge matches between PSSA, Huddersfield RC, and Altcar no longer exist. At Diggle, F-Class has ~100 names there now in its championship listing with frequent attendances by non-PSSA visitors who want to try it at longer ranges (800-1,000). FTR (mostly 308, a handful of 223s) used to be 50% plus of the field, F-Open usually now sees the larger number with various sixes, 6.5X47L, 6.5 Creedmoor, 7mm-08, 284 Win / Shehane, 6.5-284, 7mm SAUM / WSM, 338 Lap Magnum, and even a .300 RUM in regular use. McQueens at Diggle sees entries of 30-45 and hardly any will shoot 308, more 6.5s these days than the rest put together. Tactical comps are heavily attended, likewise levergun, and even Benchrest at 600 and 1,000 yards will see far more people these days than TR. Although with a factory rifle class, BR actually gets a few 308 Remington PSSs and Savage 12s, even if the winners mostly shoot 6.5X55s.
If one takes the regular bookings of Bisley lanes, and the many MoD ranges around the country for non-competitive shooting days or maybe counting as a club league round for some, 20, 25 years ago it was 90% plus 308, historic military aside which has always had a good following. So, it was a mixture of Gen 1 TR rifles in iron sights, state of the art TR rifles (iron sights) and a variety of sporters / police rifles such as Remington VS / VLS / PSS, Steyr SSGs, Tikka M55 Supersports, and suchlike, all scoped and 80 or 90% 308 Win calibre, the remainder mostly 6.5X55 and 223.
Today, you'll see a lot more custom scoped rifles built by Dolphin Guns, Neil McKillop, Norman Clark, Osprey Rifles and others, all scoped, and unless the owner wants to enter FTR matches virtually none in 308. Right now, 6.5X47 Lapua is most likely the single largest choice, but there are probably anything up to 30 different cartridges in use in four or five calibres. Then there are the more accurate factory heavy-barrel rifles from a range of manufacturers which stuck longer to 308 and 6.5X55 but are increasingly using other cartridges these days. The Ruger Precision Rifle, Sabattis and suchlike initially sold more examples in 308 than anything else, but today are more likely to be ordered in 6.5 Creedmoor.
So, to summarise, unless one is going to participate in an ICFRA discipline that specifies 223 and 308 cartridges only, there is simply no 'standard' out there anymore. factors that cause people to veer one way or another are cost (secondhand 308s are increasingly good value as the cartridge's popularity wanes); barrel life (plus for 308); precision (308 is still very good with handloads); recoil (depending on the individual's tolerance plus whether the discipline needs rapid snapshots where low recoil is a plus; external ballistics allied to the distances people want / expect to shoot over - which have become VERY much longer in the last 10 years; whether cheap factory ammo is available or handloading is going to be used for 100% of rounds through the rifle; and not least plain good old fashion. There is no definitive answer, and the advice given is often inconsistent, sometimes downright wrong, sometimes very good.
Alternatively if you are a mere mortal like the rest of us then you will be competing under UK NRA Rules, UK F Class rules, UK Gallery rifle rules, UK Target shotgun rules, UK Sporting rifle rules, or UK benchrest rules. A quick google search will locate the rulebooks for the organisation covering the target event you are interested in, you will need to find them and read them. You really do need to read them because calibers are event specific. Traditional target shooting with centerfire rifles where you shoot prone, supporting the rifle with a single point sling, only allows 308 or 223 in the UK. Most people use 308 as it's performance is better at longer ranges. If you want a 6.5 Creedmore then look at F class Open events (not F/TR) or possibly benchrest though I believe 6mm ppc is more popular. 
