The battery replacement was OK but when I was all done, the iPhone didn't connect to the cell network but was stuck on "Searching..." Wi-Fi was OK.

I noticed the comment about spring-loaded connections and oily fingers, so I took the display out again and cleaned with alcohol (the cleaning variety, not the drinking kind :) but no improvement.

So I set to Airplane Mode and back again and that cleared it ! You might find this tip useful if you have the same or a similar issue.