NCT - postponing??

The NCT on my car is technically due this week and I have an appointment for later this month. It now transpires that this date doesn't suit. Checking online, I find that, extraordinarily, the next available date at my test centre (or indeed the surrounding ones) is not until the middle of June.

Is there any issue/downsides with me pushing back the date to then?

No downside unless the car has recently been serviced and you want to get the NCT before something has a chance to go wrong. If you want a test sooner then just ring them and tell them you want to sell the car and need an NCT . . they will put you on a cancellation list.