AIBU to be sick of my manager continuing to pester me about taking holidays?

I'm with your manager on this one.

General planning of leave and cover, whilst acknowledging there are some who lack this basic managerial skill.

What the heck are you going to do with this puppy after two weeks of lurve, go back to work full time?

If you are part of a team where cover is needed or only a certain number of people can be away at any one time, then you are being hugely unfair on your colleagues to stall making a decision on dates AND expect to ring fence first dibs once you know when you want.

YABU

She'll have 8 weeks notice surely when the puppy is born? Typically notice is double the length of the leave, which I think is reasonable.

So I'd say your manager is bu unless your company's policy is stricter than this.

Is this a reverse?
Will you be ok if your manager has to decline the leave?
Can't you just cancel the leave if you don't need it? I always told staff it was better to book (then cancel if necessary) in these sorts of situations

Managers have a duty to organise work cover. They can also refuse A/L requests if they don’t fit with the needs of the business. You have to work with them here

Yabu to expect to put in your holiday dates last minute and have them approved, yes.

If I told my manager that I MAY take a few weeks leave sometime in the next few months to get a puppy, I'd fully expect the response to be that it MAY not be authorised unless booked in time for cover to be arranged and prior to anyone else booking that time off. I think your manager is trying to do you a favour, they can't just let you take time off whenever you decide for a puppy. It's not like maternity leave, there is no entitlement to time off for a puppy.

Holiday requests are not guaranteed to be granted in any job, so the longer you leave it the more likely it is someone else will nab those dates - particularly if you’re looking at a large team and potentially busy dates (Xmas, school hols etc). If you’re happy with that risk, tell your manager that. She may think she’s doing you a favour if she can see the calendar filling up.

Alternatively, she may be trying to plan workloads and not want to have to scramble last minute for cover if it’s a smaller team or she know of projects coming up. But again, if you’re happy with the risk your request may be denied, just tell her that.

can you not have a chat and explain ?

From a managers perspective:

Taking 2-3 weeks off is impactful on a business. What is your policy regarding giving notice for AL? Are you likely to give them at least 30 days before you book it or are you going to say next week I need 3 weeks off? Your manager is likely ‘pestering’ you because they feel that 1. they need time to make arrangements to cover your absence and 2. by the time you book it off they might no longer have the ability to approve it (too many people off, too much workload to distribute). Holidays need to work for both the employer and employee, dates are never guaranteed.