I have one of each that have very rough bores, I can get a barrel for each that are very nice correct,original GI barrels. I have rebarrelled many Mausers, a few win 70's and Rem 700's but never a Lee/enfield. How hard is it to pull the barrel, any tricks or things to beware of in the rebarrel/headspace process?

The No1MkIII rifle was known and marked as just the MkIII before approx 1925. But they are the same basic rifles.

The Rifle No3 is actually the P14 Enfield 303.
That was the designation given to that rifle at the same time the MkIII was given the No1 designation.
(Rifle No2 is the .22rf trainer)

No real problems I can think of in re-bbl'g a No1MkIII. I've never worked with a P14.

I've never had a problem getting the old ones off using a proper fitting action wrench and good bbl vise and blocks.
They time into position easily for the extractor cut allaignment in my experience which I admit isn't all that much in bbl switching on these.

The other normal things you do and check in a job like this apply.
Headspace can be a problem. Don't assume it's on just cause a new bbl is in place.
The No1 MkIII does not use numbered bolt heads like the No4 for ease of adj of headspace on the bolt body. A bucket of take offs is helpful but not usually the normal thing hanging around the shop.
Check the bolt lug engagement. Often one has little or no abutment. The bolt head should wind in to the body to a spec and not over draw too much. Firing pin protrusion checked and adjusted, ect.

Assuming you mean Mark 111 and not Number 3, and Number 4, some of the barrels are quite tight as the shoulder is large. I always part them off; no danger of warping the receiver that way.
Oh, done lots of actual Number 3s, we all call P14s; same thing.

I corrected the header, between model #'s and Mark #'s I always get screwed up! These are Lee/Enfields #1 Mk3, and #4 Mk 1. I have a good barrel vise, but would probably have to make a action wrench as my old style Mauser type would be oversized. It seems that these barrels may be over- torqued in like the P-14's?

I have a .303 reamer you can use if needed.
I wondered what a tight chambered .303 mk 111 would be like. Had a like new P14 barrel I cut and fitted. A nice tight chamber does wonders in the old SMLE!

+1 what dpcd said. I just removed the barrel from a number 4 #1 and had to make a parting cut to relieve the pressure.. It still popped pretty good when it broke loose. I made an action wrench to fit the contour of the action at the receiver ring out of 1/2inch plate.

US small arms are Models; Brtitish are Numbers and Marks,
Like our Navy which also has Mods (modifications). Very uniform system of nomenclature; not.

Theback40, thanks for the offer, I may just take you up on it! Thanks to all for the info, I will need to build an action wrench and get the barrels.