Not sure about tenses, but you will have to have ways to explain to people
You don't know me yet, I meet you in your subjected future 2 years from now.
In 6 years absolute from now, I will have been going to go to the Cretaceous for 3 years subjective.
Your best bet is to reread some of the classic time loop stories. (All you zombies, By his bootstraps, The Big Time (Fritz Leiber) ) and make a list of the types of situations. Reduce them to diagrams, and then invent tenses.
I think you will need past/future absolute/subjective correlated/disjoint
The first two I think are obvious correlated: subject and object are on the same or close to the same time path. disjoint: One of you is out of sync. If I drive the time machine correlated present. "I went to 1850 to steal the Hope Diamond" steal is past disjoint.
You will have terms for 'self consistent causality loop' (Because of A, then A) for dual mode causality loop (Because of A then B, because of B, then A) multi mode loop...
Depending on your rules you will have principles like 'conservation of historical change' 'collapse of minor universe splits' (this explains both the disappearance and recovery of missing socks and why you didn't see your car keys the last 4 times you looked on the hook, but they are there now.)
Anytime you find yourself drawing pictures to keep track of things then generalize it and invent a word.
I think you need tenses for personal past that differentiate from absolute past. "I drank a milkshake yesterday:" personal. "I drankah milkshake yesterday:" historical. "I will drinkah milkshake yesterday:" Personal future, absolute past.
In this case I think that the auxiliary verbs in English add confusion. May want to look at pigeon languages.
English is almost entirely positional. We aren't aware of 'case' of nouns. In declined languages you become very aware of case. AI research in computer understanding of natural language is re-inventing case as a way of extracting meaning. We are aware of case intuitively: We routinely join nouns that share case. "Mom is baking." "Susan is baking" => Mom and Susan are baking.
"Mom is baking" "The Pie is baking" !> "Mom and the Pie are baking" Mom is the subject, doing the action. Pie, is receiving the action, so is the object.
Take a look at https://en.wikipedia.org/wiki/Grammatical_case
Imagine trying to explain music without the words for chord, major, augmented, diminished, modulation, tonic,
You probably don't want to go with a full jargon that professionals would use. They have had decades to arrive at it. Just a sprinkle.
There will be other conventions:
If there are multiple time points of an individual present, you always start introductions from the most senior down.
You never mention a person's death to that person.