This is not a problem with the glow plug relay. Even with a totally bad relay, or even a disconnected relay, the engine should still turn. And if you crank long enough (30-60 seconds) with healthy compression and mild ambient temps the engine should still start.

However, since you can repeat the process, it seems to be related in some way. I'd suspect a weak battery. When the glow plugs are lit it doesn't have enough juice to turn the starter. Only after the relay is no longer drawing on the battery does it have the power needed.

That's an easy place to start anyway. You can have it tested.

If it's not that, your cable connection to the starter may be dirty/corroded, same with your battery ground cable. Or you might have a faulty NSS or starter.

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