If this doesn't work then I will go to the other link that was
This is good, and I think it is what you want. You are speaking about your current plan.
If that doesn't work then I would go to the other link that has been provided.
This is a valid sentence, but I don't think this is what you are trying to do. In this use, you have just given advice to a listener about how to do something, and your listener is going to make an attempt at it. You are giving him advice about what to do if that advice does not work, by specifying what you would do in that situation.
We could say "If this..." here, instead of "If that..." - the difference is in how close the speaker feels to the strategy being suggested, if you follow my meaning. If both the speaker and the listener are working on the problem closely together, use "this". Using "that" puts some distance between the approach being suggested and the speaker. In many cases, they are interchangeable, but if the listener is going home to try the suggested approach, or waiting until later to do it, use "that".
If that hadn't worked, then I would have gone to the other link that was provided.
This is correct, and the speaker is narrating his thought process during a situation that has happened in the past.
There are some degrees of freedom in all of these, but I'll point out a couple in the most recent case:
If that didn't work, then I would have gone to the other link that was provided.
is functionally the same (with a subtle distinction of bringing the conditional statement more into the present), similarly to how
If that hadn't worked, then I would have gone to the other link that had been provided
is fair, but perhaps makes an odd, very vague distinction about when the company provided the link. I imagine some speakers and some rule systems will have definite ideas about what is best in some of these, but the most important thing is to get the tense right in the main part of the sentence - e.g. "then I would have gone", - and smoothing over any conflicts with your "this"/"that" choice and related elements of the sentence.
And one more thing - you could simplify some problems by just saying "the link provided" in all the above cases.