Removing Grease Nipple

Use a vice grips. Squeeze it tight at the butt of the nipple and twist it off. Have you freed the ball bearing at the top of the nipple and seen if you can grease it after that.

Usually the nipple is threaded. And it should twist off. However it looks very rounded.

What usually blocks them is a bit of hard grease below the ball.

You have a better chance trying to twist out than trying to pull out. The twisting action might loosen it.

I be reluctant to drill in case filings got down into the grease. Brute strength and ignorance try a small cold chisel to pop it out. Leave it very hard. If it lose it should pull out by brute force

I would be more inclined to cut it off with an angle grinder and see what I would do then have you tried taking the housing holding the nipple off. That looks like a bearing at the bottom with a treaded compression ring holding it in place. See if that will twist off