If you use Trimble Access and use an FXL file you can just add date, time, or both as attributes for whatever feature codes you want to see them on. You do this in Feature Definition Manager.
You can click on the arrow under the Export button and open Export Format Editor; you can export anything that TBC has in its database to a custom format. This includes Timestamp on total station observations and Start or End Time on GNSS observations. That editor is pretty straightforward once you get into it.
TBC will create Vector Spreadsheets and Optical Spreadsheets; depending on the version you have these commands are probably in the Survey tab. Spreadsheets can be edited for content in Project Settings > View; you can also drag the columns to put them in the order you want them in. Once a spreadsheet looks the way you want it, you can right click in the upper left hand corner, hit Select All, right-click again and hit Copy, and paste it into Excel.
ASCII File Generator is a utility that you can get from Trimble’s web site on the Trimble Access support page. They have all kinds of report formats available, and there are descriptions of them on the download page. ASCII File Generator can also be called in TBC by going to Reports > Job Report Generator.