Hi Stockwell
I am not sure if I understood your question well. But if you are confused with the sling resolution The Basics this will be a good read for you.
Let me try to explain what might have gone wrong in your case
Let’s say you are creating a page with /apps/uc-base/components/templates/global template, Now the first thing sling will look for is a page named global.html. If it is not present then it will check for the parent (sling:resourceSuperType)
Here, your parent page, wcm/foundation/components/page, when invoked will look for page.html (assuming you don't have any specific extensions in your URL. Lets keep it simple
) under its folder name and not the child. It will look under child only when the page.html have an include and if that file is present in the child with same name. For eg:- In the below screenshot wcm/foundation/components/page , the page.html
at line 20 has a include to body.html. Sling will first look for this file under /apps/uc-base/components/templates/global, if it is not present then as per precedence it will look under wcm/foundation/components/page
So, to solve your problem, the pieces you want to use from the base page, should be included as a body.html under your /apps/uc-base/components/templates/global so that your resolution picks up this file while running.
PS:- Its not always mandatory you have to add body.html only, but to make sure the foundation page is included in your structure, make sure your base page starts with body.html and further whatever you want to include just keep adding on that . Sling resolution is a big concept which cannot be explained here with due respect to all its concepts.
I am sorry if I confuse you more
I think once you have a better understanding on Sling resolution this would be a cake walk
. Below are some good reads
Apache Sling - URL decomposition
Sling Cheatsheet - docs.adobe.com
The concepts remain same whether in 5.6 or in 6.3
Hope this helps and too much of explanation has not confused you 
Thanks
Veena