Hi Tom, congratulations to your new home! You can put some glass or wooden shelves in the alcove (for the plates, glasses etc.) with hidden LED strip lights or turn it into a display cabinet if you are a collector. Another idea is to put a minibar in it; or if you like drinking coffee after lunch and dinner you can create a small coffee bar there. Above the alcove hang an artwork on the wall or use other paint colour to make it a feature wall.

I've put decorative logs in mine.

You could put a faux wood stove, with a faux flue pipe in there.

A plant; group of lanterns with candles; logs; or go wacky with a fish tank ;)

Wine rack?

We had all our awkward oversized children's books in ours at previous house. So you could fit a lovely wooden shelf and have 2 lots of books in that space.

Here’s some ideas from Pinterest

A bioethanol stove?
Some look just like a traditional woodburner, others more contemporary. The advantages for your dining room is that they’re freestanding - no venting required - make no mess, and are very controllable. They won’t put out as much heat as your real log burner, though.