I recently had a bit of a nightmare with this problem when building my car back up following restoration. My problem was fuel supply, not enough fuel pressure.

So I'm thinking its to do with fueling. Cold it will use the cold start injector to help it fire, and will run on this for a couple seconds before using the 4 Injectors. Fuel pressure should be 80-90psi between filter and metering head. The cold start will work on less than this, but your mechanical injeftors need this pressure to work correctly. Your symptoms sound like a lack of pressure.

I've fitted a fuel pressure gauge between my filter and metering head and it's done wonders for problem solving!

The system should also hold about 30psi when turned off. If it's not holding pressure, I.e leak in fuel line or one way valve at pump failing then it won't be happy turning back on hot.

Like somebody has suggested, check to see if your plugs are wet when trying to hit start, this will tell you if fuel is coming through injectors correctly to eliminate a fuel problem - but my hunch is its to do with fuel