If you ping the links you've built, generally see SERP movement within a week or two.

Generally after I change a site structure or do a redesign, some URL's are broken and not working. In those situations, you need to do the following:

-301 the broken URLs to new URLs
-Take an accurate snapshot of where your link building efforts are pointing
-Make sure you don't mess up a page that has tons of highly authoritative links

After you do that and change your structure, it's a great idea to build links to your NEW structure to build back any lost authority and start ramping up those ranking efforts.

Does that make sense?

I just built a brand new website (brand new domain), built it, SEO'd it, and built links and it;s ranking page 1 / page 2 for a medical term that gets 40,000+ exact searches a month. Only been 8 weeks. If that gives you any indication.

I'd say give it 3-4 weeks, then start building links and/or looking at your SEO strategy.

When choosing a term to rank for, in my experience, choosing a term that has low competition means finding sites that rank top 10 that do not have high domain/page authority scores, relatively low moztrust/rank, and of course really low backlinks. If a site has 10 backlinks, as long as they're from sites that are under 50 DA, they're easy to rank for.

The reason you get different results from KW Difficulty Tool using quotation marks is because that tool pulls the top 10 google results for whatever term you put in. Generally, the top results are different when using quotations. I recommend you NOT use quotation marks when using the KW difficult tool.

I would do your own research and optimization and stay away from using someone elses work/data, to me, it seems prone for errors and miscalculations which lead to loss of money, time, and other potential fallout.