Given that I have a lot of experience studying Jones genealogy (i.e. my surname), I took a quick look at your Jones line and David Bowie's Jones line.
Your Jones line goes back to your great-grandfather, William Jones - born in Arkansas in 1864 (and apparently back to Elbert Jones - born in 1844?). David Bowie's Jones line is from Yorkshire, England going back to his gr-gr-gr-grandfather, Joseph Jones - born in 1790.
So - unless you find that your Elbert Jones (or his Jones father or grandfather) came from Yorkshire, England, it does not seem likely that you share a recent Jones ancestor with David Bowie. If you want to check further, you should look into Elbert Jones, since you list that he lived until 1944. As such, there should be many records on him. But - unless you see a Jones from Yorkshire within the last 5 generations (i.e. if Elbert, his father or grandfather came from Yorkshire), it does not seem likely that you share a recent Jones ancestor with David Bowie.
Looking at your possible Oliver connection, your great-great-grandfather, Richard Oliver (1825-1920) was from Van Buren, Arkansas. David Bowie apparently has one Oliver ancestor, a paternal great-great-grandmother, Mary Oliver, who was born in Yorkshire, England in 1827. Once again, you could try and trace back your Oliver line to see if Richard Oliver's father or grandfather came from Yorkshire, England. But, again, it does not seem likely that you share a recent Oliver ancestor with David Bowie.
To be perfectly honest, there is nothing in your tree to suggest that you have a recent ancestor within the past 3 or 4 generations who came from England. As such, the logical conclusion to draw at this point is that you probably do not share a recent common ancestor with David Bowie, based on the simple fact that his entire tree is made up of ancestors from England going back several generations.
The only other suggestion I can make is to keep filling in your family tree to check for sure that you don't have any ancestors from the past 4 to 6 generations who came from England. Again, it doesn't seem likely, but you will only know for sure if you fill in your entire tree going back 4 to 6 generations.