98 A3 1.8T - Turning over but not starting. Intermittent. Please Help...

Since last week my 98 A3 1.8t sometimes won't start. It turns over as if there's no fuel or it's immobilised.

It doesn't have the flip remote, just 2 keys that work central locking.

It makes no difference what key is used.
Theres plenty of fuel in the tank.
It did it today and I held the key turned clockwise in drivers door which wound the windows down. It started after this, but is sat at home now and will not start whatever I do.
There is no immobiliser warning light on.
It's most likely to happen when its been parked between 30 mins and an hour. After parking overnight it always starts.
Sometimes disconnecting the battery for a few minutes helps.
Most times it will start after being left half an hour or so.
I can hear the fuel pump working.
The second from left fusible link on the fuse box over the battery was slightly melted, so I replaced the box.

Any ideas anyone has would be greatly appreciated as we are scared to take it out in case we get stranded.
I'm in New Zealand and it's a Jap spec model.

Thanks for any help you can give.

Thanks for your reply Squirt. I'm getting it booked in today to have the fault codes checked. The problem does seem to be when it's hot or even just warm. I've just tried it now, after being parked overnight and it starts fine. I can be pretty sure if I drive down to shops and try to start after half an hour it won't go.

So it seems it was the crank sensor. That's what the garage diagnosed and replaced, so fingers crossed thats got it sorted. Thanks for the advice.

Thanks for that. Hopefully this will be the end of the regular trips to the garage. New front shocks, lower arm mounts, rear brake discs and pads, thermostat, temp sensor, fuse box and crank sensor all in 2 months.