Hi.
So your neighbour has already extended but doesn't want you to do the same? Did I get that right?
How cheeky! If you have been given planning permission to go ahead, then you may do so. But you have to get that agreement signed. You need to instruct a surveyor. It may get expensive, but the fact that they already have extended their home will mean that their surveyor will probably point this out to them. You are not setting a precedence in the street, if that is correct.
We have been in this position too, and we did lose the relationship with one neighbour. But we got the extension we were entitled to. Regardless of neighbours jealous rants.
Take photos of their side and yours. Log everything on paper. That way they will have the assurance that you are taking their worries seriously.
Our neighbour wanted us to make our extension much smaller as it would suit her. But it was not her house that was being extended and she wasn't the one paying for it. Or living in it.
We did everything we could to make the time easier. For both sides. we fixed anything that went wrong, bit nothing major did. It's a very stressful time, but it was worth every penny and now our neighbour has left. So if we had danced to her tune, we'd have been left with an extension we didn't really like and were left with whilst she has moved on.
Good luck.