Link to it from a page Google already has
Googlebot walks your site the way a person would, by following links. A new page reachable in two or three clicks from your home page gets found because the pages above it get crawled regularly and it's sitting right there in them.
A page nothing links to, sitting in the sitemap alone, is the harder case. Google has the address and no signal that the page matters, and that's the shape most often reported as found but never fetched.
So when you add a town, add it to the list of towns a person can browse, and link it from the towns next to it. It takes a minute and it's the part with a real mechanism behind it.
The sitemap, and where it goes
A sitemap is a file listing every page on your site. Put its address in your robots.txt file and submit it once in Google Search Console. That's the whole job, and once it's done you don't resubmit it every time you add a page.
The old trick of pinging Google to say the sitemap changed doesn't work anymore. Google retired that endpoint in 2023 and said so in writing. If a tool in your stack still does it, it's talking to nothing.
Time, and what nobody can tell you
A brand new site with no history takes longer than an established one, and there's no published number for how long. Google doesn't give one, and any company that gives you a date made it up.
What you can do is watch instead of guess. Search Console's page indexing report tells you which pages Google has, which it has seen and skipped, and which it hasn't fetched. That report is the answer to the question, updated for your own site.
The shortcuts, and what they really do
Google's Indexing API only supports job postings and livestream pages. Sending it your service pages returns a success message that means the request arrived, not that anything got indexed. Services selling instant indexing are selling that receipt.
The Request Indexing button in Search Console is fine for one page you just fixed. It isn't a way to push a few hundred pages, it confers no importance on a page, and using it in bulk changes nothing about what Google decides to keep.