I agree with FlyingZebra about trying to start with a rifle with the correct bore (caliber) and twist rate. I had two No.1 conversions that took years and two different smiths to get them right. Other important considerations IF you go with a new barrel:
1- possible supply chain disruptions (my custom revolver build was modified because builder couldn't acquire material for a new barrel or cylinder),.. may not be the same for you, but when you find a builder, he'll let you know "then"
2- it's better if the barrel contour of new barrel match the donor stock and sights you want to use. Don't expect a new barrel to work with your existing donor gun unless you're guaranteed it will fit. Determine that in advance because, otherwise, it will add complication, time, expense, and some frustration. Been there, done that!
3- don't take your open / iron sights for granted. This can't be taken for granted for two reasons: barrel contour and available rear and front sight height combinations for the cartridge and load you want to shoot. Regulating sight height for my new cartridge ended up being my biggest problem because my donor gun (a 45-70) had a factory setup and barrel contour that just was not going to work with my new cartridge.
So,.. start with a donor gun with caliber and twist rate that is already very close to what you're changing to. (This is good general advice for any conversion.). Determine how you want to sight it. If all you want is a scope, then this will be much easier. The No.1 is short and can be well-balanced, so determine barrel length and changes to the donor gun so the end result balances well.
EDIT: by relationship between sights and barrel contour, I was referring to available sights being adapted to the new contour without excessive modification to fit that contour shape. NECG offers the best, IMO. Everything is possible, but my point is to take care with choices to keep cost and build time down. Plenty of smiths will take on a job, but take two years to complete it, and sometimes it's all the little stuff that can take up all the extra time.