How to re-install updates to Google system apps?
I'm assuming your phone isn't rooted. If you know exactly what build/version you're after then do a quick Google search for 'Google Play Services XXX.apk' and manually install the later version over the top of your current one, and the same for the next one. They are available on aptoide, blackmart and other alternate stores.  

If you get any packages error, or system error, I would suggest just do a reformat and reinstall all your content- it's generally quicker than scowering the net for the cause behind the issue.