All you would need would be a "neck" reamer.
You can get one out of Ukraine made up and delivered to your door for around $85C.
Just Google Ukraine reamer builders.
The same folks are on Ebay and they sell pistol caliber chamber reamer kits, which include rifling spuds, for around twice that much.
The UK vendor number is xz2010(807) and has a good reputation.
Any smith that has a 303 Brit, 7.62x54r, 7.62x39 neck and throat reamer should be able to do the job you require.
The only other way to open up that neck properly, would be to remove the barrel if it isn't already off, cut the chambere with a normal reamer and use an inside turning tool to open the neck a few thousandths.
Opening the neck only a few thou can be tricky, especially without a pioted reamer. There are good machinists around most areas that could do the job for you, without a reamer.
I saw a .380acp chamber cut into a barrel blank, without a reamer. The 380acp case only has a couple of thou taper. That chamber was every bit as good as one done with a reamer.
Rotation speed was very fast to keep things very smooth and the cuts were only a few thou deep per pass to stop deflection on the very small inside cutting tool bar.
The throat was cut with the same tool, but it had been ground with an angle for the bullet to feed into the leade.
That little pistol went from prohib to restricted, with a 110mm bbl and actually shoots better now than it did with its original bbl.
Any good machinist should be able to do something similar for you.
In your situation, I wouldn't worry about the throat at all. The tighter the better IMHO
The thing is, are you doing this with a semi auto or bolt action??