Once the magazine is set up correctly, the SMLE is about as slick as you can go with a bolt gun.
I suggest some of you brace yourselves now as I'm about to reduce some sacred milsurp cows to hamburger. When it comes to shooting milsurps, the biggest factor you will notice is the effectiveness (or not) sights. My observations.
Mauser 95s and 98s. I don't know who designed these sights, but I doubt he/she ever actually tried shooting them in realistic conditions. The barleycorn front sight combined with the very shallow notch in the rear, barrel mounted, sight are simply awful in tricky light. In certain light it's almost like the front sight moves. Maybe it's me, but look at the sight picture of a 98, then pick up a Swede 96. The Swede has vastly superior sights.
SMLE. The straight-sided front sight is great, but again the rear sight notch is very shallow and difficult to pick up quickly. Negates some of the advantage of the slick bolt, I reckon.
Springfield 1903. I only have an 03A3, not a WWI veteran. The receiver peep on 03A3 is great, but the very thin front sight is difficult to pick up when the background is cluttered (real world) but fine on the range. What is that thing about the Springfield being the best target rifle?

The bolt on my 03A3 is very slick, almost as good as an SMLE.
Carcano. Yes, seriously. The front sight is almost straight-sided, and it has a deep and wide rear V notch for quick acquisition. It works well for me, but I've read grumbles that the sights limit ultimate accuracy. How much of that is truth, down to poor ammo, or a "not invented here" attitude to the sight picture I cannot say. The sticky action of the Mannlicher bolt spoils the Carcano somewhat. Even my almost unused M1941 requires a little wrestling compared to a Mauser.
Mosin Nagant. Problem here is "which model". Nearly all the M91 rifles we see here have a Mauserish front sight and a rear sight with a fairly sizeable V notch. Sight picture is so-so. The early 91/30 rifles have a front post and a rear sight with a reasonably sized U notch. They work well. Unfortunately, when WWII got going, the Soviets started using a Mausereque shallow rear V notch, probably because it was quicker to produce. It's hopeless in poor light. Mosin bolts vary greatly in smoothness. Mid WWII Soviet rifles can be as obstructive as a bad Carcano. A Finnish modified gun can be very slick.
Berthier. Huge, wide front sight. It should say "Oversize Load" on the back. The rear sight is set up to match. Camp Perry boys from the '20s would have a fit of the vapors at the sight of one. It's a combat sight, plain and simple.
You put the target in the middle of the sight, a complete reversal to the US philosophy. THis guy explains it better than me.
Perspectives on French Rifle Sight Design The Berthier bolt is a bit clunky. If you think it feels large, that's because it is. The French 8mm round has a bigger case head than 30-06 or 8mm Mauser, so the bolt is chunkier.
Enfield P14/US Model of 1917. Brilliant sights. Nothing to add on those. Biggest issue with the bolt is that people screw with them by fitting a kit that changes the operation to cock-on-opening from the original cock-on-closing. Vandals. Either way, they are nowhere near as slick as a SMLE. The Winchester P14s can be ammo finicky, as it seems that Winchester had their own ideas on the precise geometry of certain areas of the receiver. Many P14 parts are not interchangeable between the ERA and Remington rifles, and those made by Winchester. I think that issue was fixed on the M1917.
Enfield No.4 Mk1 and derivatives. The rifles with the vernier rear sights are great. The peep is a nice size, and the vertical adjustment fine enough for accurate fire. The two position 300/600 yard flip sight is OK for the intended purpose in wartime.