All the straight-pull rifles seen in the CSR video will have been AR-15 based and likely also all .223s. (There are a few 6.8s and others, but 223 is the overwhelming choice for the platform.) Under UK law, the rifles cannot be converted semi-auto or selective fire weapons and have had to be designed and manufactured from scratch for manual operation. So there is a lot of mix and match with US supplied components, some machining, and a bit of use of custom made parts depending on the builder, his views, and his capabilities / facilities. Many builders take US made off the shelf 'uppers' that best suit their needs and machine them to accept operating handles, one has both them cast for him to his specs and does final machining in his workshop. Bolt carriers may be modified standard, custom made, or manufactured in-house. Barrels are as per US norms but are not drilled for a gas vent. Bolt gas rings are not used and all other gas operating parts are omitted, although you well still find the original carrier gas key in place solely to act as guide. Lowers are basically as per US semi-auto norm without any alterations although there should be a paper trail to show that the receiver was not manufactured to be part of a semi auto rifle.
There are four ways of operating the mechanism.
The oldest and least satisfactory is to use the original AR T-bar charging handle with or without bits added on to move the grasping ball or bar forward nearer to the trigger hand.
One is to mill a slot in the left hand side of the upper receiver and have a captive cocking / operating handle installed therein. A peg on the handle contacts the left front edge of the carrier. The main manufacturer of this system uses an FAL type folding handle. (Southern Gun Company)
The third is to mill a slot on the right hand side (actually can be either side) and to attach a much longer and sturdier handle to the carrier side using two bolts. This allows a lot more hand purchase and for much more pressure to be applied directly to the carrier - important with the AR design having no primary extraction cams and often suffering hard bolt opening with high pressure loads. The handle has to be removed for field stripping, but that's not onerous, a minute with an Allen key - just don't drop and lose the bolts if you're doing it outside behind the firing line.
All three of the above methods see the rifle mainspring retained so that hand operation is solely to open the bolt, extract and eject the fired case, reset the trigger and also to move the bolt assembly back through its full normal amount of movement in the upper receiver to either be locked open by empty magazine operation, or released at the end of its rearward travel for the normal AR operating cycle chambering a new cartridge etc to take place. Methods two and three are sometimes combined to give ambidextrous operation. In the video, everybody seen close-up uses their trigger hand and moves it between the trigger and operating handle. With practicve, this can be done VERY fast and with hardly any disturbance to the aim. With milder 223 loads, the right-hand can stay on the pistol-grip and the left used with the less powerful captive operating handle.
The final and much more controversial operating method is Southern Gun's 'Lever Release' system which is half way between semi-auto and manual. Gas operation kicks the bolt open as in a semi-auto AR but it is locked back until a thumb operated lever on the lower receiver is pushed to release it. It was developed for pistol calibre AR-type 'gallery rifles' used in UK Bianchi / 1500 / Police Pistol type shooting disciplines and competitions where short barrel carbines have replaced pistols and revolvers which are also 'Section 5 prohibited weapons' here, but SGC's Bob Clark has developed a 223 version. I'm not in he know as to how many, if any, of these have been made / sold. I'm not a High-Power or CSR competitor, but I don't think any are used in these disciplines.
The systems / methods have been applied to AR-10s with cartridges like 308, 260 and 6.5mm Hornady Creedmoor. (the same also applies to cartridges like the 6 and 6.5 Grendel in AR-15s.) They don't work very well due to the primary extraction issue and in fact have to be severley downloaded from bolt-action loads to work at all. Various operating handle bolt-ons have been made and tried to provide this, handles that swing through an arc working on a cam initially to do this job before hitting a stop to then pull straight back. The AR-10 is dead basically in this role, but there are sales of redesigned US made LMT rifles that apparently work at the very least reasonably well, for all I know very well, as straight-pulls. However for serious competition, I believe that bolt-actions are still preferred with the larger cartridges, custom built jobs, Accuracy International etc. AR-10s are also usually regarded as too heavy and cumbersome for CSR.
Re MTM's point as to White Oak Armaments' views on AR mechanisms being unable to withstand full pressures if the gas system is disconnected, I've not heard of that. The 15s use absolutely full-power European Nato ammunition loaded to 61,000 psi. This may be the case with hot larger cartridges in the AR-10s, but you download to get the damned action open without having to use both hands on the operating handle. (You think I'm joking don't you, but I'm deadly serious!) I can't comment on the LMT rifles, but if they can't shoot standard Nato ammo they're not of much use to many would-be buyers. We're also seeing straight pull SVD Dragunovs in 7.62X54R, FN FALs and M14s/M1As in 7.62 Nato, and the odd .30-06 M1 Garand etc and I've never heard of anything breaking or wearing out fast.