That's a very nice question.
In general, we always prioritise user experience.
Flights search engines could deliver results instantly but they have a longer loading as users feel more secure that they have done a thorough search.
When it comes to 2x times slower.
2x is probably ok when it is a couple of seconds and probably is terrible if it is for years. It all depends on the context of your users. If the kind of work your site is a life or death matter then faster is better, if your users are more relaxed with the time it takes and it is more important for them to know when to check back then telling them the expected time would be the best option.
Make it faster when
- Loading time is proven to negatively impact adoption, use, reliability, trust, etc.
- Speed is a matter of life or death
- Speed is a matter of making money (e.g. clients on the phone need urgent answer)
Show the loading bar if
- Multiplying the time by two doesn't create a time that would put
users completely off
- You users can afford to wait while probably doing something else
More things to take into consideration
Maybe you need a hybrid solution that for some processes you get the loading bar and for some, you don't.
Maybe you need alternative options: e.g. showing loading bar and tell users that by hiding it they can speed up the task, asking users if they want to receive an email notification when the task is ready etc.
In any case, the best idea would be to see what users would prefer according to their context and needs.
Good luck with it though, it is a very interesting challenge. Do post back what decision you picked in the end. :)