First of all, 3 links won't make a big impact if they are on a big domain build on a completely new page.
Here Is an example of web2.0 that we all make they are all big domains but when we make a new account and subdomain we are starting from scratch so to speak on a property that allows you to push it up fast as the main domain is old and well developed with millions and millions of links already pointing toward it so when you build T1 link on property like that you are suppose to build a bunch of T2 and T3 links that will push your T1 page to build juice for it and pass that juice to your money site.
When you are buying a backlink on an aged domain that has info as:
DA60 DR60 RD1000+ Backlink30,000+
If you want to take advantage of that domains development you have to buy a link on the 1st page usually or any other well-developed page of that website to get the most out of it and those backlinks cost a lot.
About the time it takes for the results to be shown.
Usually, it goes like this:
After I finish one backlinks campaign the keywords that I marketing in the campaign starts moving a bit up after 2 days and after that on going for 2 months they go up down from day to day.
Some links start showing in ahrefs after few hours (not many) and continue showing each and every day (gradually).
After 14 days 10-20% of links show up in google console as a first wave.
2nd wave of links shows up after 28-30 days.
After 2 months I can see the final results (more or less) of the work that I did that day with that campaign as around that time keywords that I was working on in that campaign stabilize at some position.
After a month of tracking if I don't like the position they are moving around I would push another campaign to push them furder.
All in all this process continues and repeats itself as we are not the only one that is fighting for those keywords. Depending on the amount of competition and how much are they working on it can take from a few months to few years to reach the top spot.
Remember that SEO is a marathon and PPC is an expensive sprint.
PS: Idk why are you so much focused on the DR of your website as Google doesn't care much about DR. You can see so many small DR websites ranked better than big ones that have huge DR because the small guy is focusing on a specific keyword ranking and working his but for that keyword to get it ranked out will the big guys with big pockets are doing some other game.

So forget about DR and focus on your keywords to rank them high and get traffic and to turn that traffic into customers because at the end of the day the only thing that is important is how much did it end up in our pocket.
Best of luck,
danedral