1950 ES2 running issues

Exactly what work was done in this restoration? Did you do the work yourself or someone else?
Does the tank have a filter? Have you checked the flow from the tank itself? Tank been lined? liner coming adrift? Has the engine been apart? Was the valve timing checked with a degree disc?. It is not unknown that the timing marks can be wrong. Lots of things to check. 

You need a theoretical 132 ml/minute fuel flow for a 25 bhp bike.  More than that is good, as a safety margin.

I don't know know what exactly you unscrew or open, to let petrol flow through a pre-Monobloc float bowl.

If you think it's fuel system, it's probably electrical.  As Peter says plenty to check, but my tuppence is on the mag.

Hi Howard,

   I am in agreement with Jon that the mag is the problem. If you have a long hill near you, try the bike up and down and see if the misfire happens at the same speed. If it's fuel starvation when you go down hill the effect will be much less than when you go up. Most likely the mag is failing when it reaches a certain speed, possibly the points.

What is main jet size?  And has it come loose?  Is the air slide fully up and clear of the throttle mouth?  If it's only half way up, it won't have any effect until the throttle slide is also half way up.

If the ignition system (mag or coil based) is only just managing to produce a spark then changes in the throttle applied can result in the spark failing to jump the plug gap. It is known that if there is more fuel in the combustion chamber the spark finds it harder to jump. As a consequence you get a missfire and a black sooty plug and exhaust because of the unburnt fuel.

A magneto usually gives better spark when spinning fast.

Things that can make it fail to spark when pulling hard are fouled plug, plug gap too big, high resistance (such as suppressors) in the HT circuit, dirty slip ring (the slip ring is not the cam ring).

If it's rpm related, rather than throttle opening related, could be points stiff to move on the pivot, or points pusher stiff to slide, depending what sort of magneto it has.

The HT lead should be copper cored.

Hi, Another advocate for getting the mag checked. The wider the throttle is opened, the greater amount of energy required to make the spark jump the plug gap. As a quick check, if you can fix the ht lead such that its core is around 1/4" from the head, you should be able to get a spark to jump across at a good kickover speed. (Plug out!) 
If no joy, make the gap smaller and recheck, but a good mag should easily jump a 1/4".
Regards, George 

..and foolish (as opposed to being old and foolish now) I bought an old Norton single that had reposed in somebody's front garden (remember when we could do that!) for a couple of years. I eventually got it to start and run, but it cut out every time on opening up the throttle. One of the roads into our estate (council, not private!) had a fairly long, straight hill. Managed to get up to the top, turned round and set off back down hill. 1st, 2nd, 3rd, still low throttle, then rammed it into first. (I think I looked behind me, but probably not). An impressive skid, but the engine, now forced into a higher RPM than previously, decided not to explode but actually began to run a lot better. Was fine(ish) for quite a while afterwards. Never really sussed what the problem was!
As they say on the telly, don't try this at home kids
 

Blocked petrol tank cap hole?

I still think it's worth checking if the choke slide is not opening fully.  That would account for bogging down at wide throttle opening, and a sooty plug.

It still sounds like the main jet is too large.
If the engine actually stops when you go over 1/2 throttle, then going down one jet size won't make any difference. You will need to go down 3 or 4 sizes as an experiment. 

Has it got an air cleaner? Perhaps remove it completely and see if that helps.

I'd also replace that HT lead with a copper one. 

As David Cooper says, a magneto usually works better as the revs increase. 

 I had a BSA M20 with a failing magneto. I eventually got the plug gap to 10 thou to keep it going, but once started it ran better the faster it ran.

Let us know what fixed it when the time comes.

Don Anson 
Melbourne. 

tension the cable to open the choke... is correct....

Partly because I find this post very interesting and want to receive updates (which you now automatically get when participating in a post). Partly because your initial question raised a question for me: How do you cut the engine if not using the valve lifter? I've got an MO1L and see no other way than using the valve lifter to cut the engine.

As for your problem: I've had similar problems, but with bikes that featured an ignition coil. When the temperature got up, a break in the windings showed up, displaying exactly the problems that you're describing, although they were more rev related.

Could it be the windings in the magneto? You might be able to borrow one from someone?

As Tim D. has pointed out, if its the mag at fault then it should not be throttle related but engine speed related, has the OP tried to find the fault while working through the gears?

Also, cruising at 60 mph with only 300 miles on the engine?  I'm assuming a total engine rebuild, new bearings, top-end, rebore and so on.   60 is too fast for a "new engine" with only 300miles on it, the poor thing will be working quiet hard at 60mph. 

40-55mph on a small throttle opening, in top gear, would be plenty fast enough at this point in running in.