A few things.
First, yes neighbours can object and will be taken seriously.
But neighbours don't make the decision. That will turn on your local authority's tree policy (if they have one) and on the choices made by their Tree Officer.
You need to decide how much you care. Are you sure it's damaging your foundations or just worried?
If you are sure - or fairly sure - you may want to submit a surveyors report with your application.
You may also need a tree expert. I can link you to the one I've used if you want.
It can get expensive (mine did).
So, if you're not THAT bothered you can just fill in the form yourself. But do it carefully - remember you're trying to persuade them it's a good idea.
Also it can help if you offer to replant a really nice tree.
I actually spoke to the council Tree Officer, several times. tbh I'm not sure it helped and it might even have done the opposite!
Finally, don't believe what you read on Mumsnet, including (maybe especially) from me. This is a very technical and complex subject. Good luck.