At the moment there is no simple way to see when the next expert will arrive on a base.
If you are familiar with the mechanic, and can work out how long all of your buildings have been working, then you can do the maths to get a good idea. However, if you have had a varied efficiency, unknown idle time, or demolished buildings, then you can’t reliably work it out for yourself. Similarly, new players won’t know at all until they find the table of progress towards experts in the handbook.
My suggestion is to add a progress meter on each category of the Experts buffer showing the percent “completion” for the next expert in that category.
The header over the progress bar could be something like Recruitment Progress or Expert Training, with a tooltip along the lines of “Making products in buildings of a given field will earn you progress towards your next expert in that field.”
It would be a small improvement but I believe it would be helpful in making this mechanic (and your progress with it) readily evident to new players. It would also add another data point to longer established players who want to calculate more exactly when their efficiency rates will tick over and they will get to update their spreadsheets 
I’m aware that with Populous coming it might be a bad time to suggest this feature, but if progress towards experts remains a similar mechanic, I think it would be a neat way to “show, don’t tell” a small aspect of the game.
Thank you for all your work on the game, I’m having great fun with it!