I feel like this is another sadly underrated (and under-developed) relationship that had so much potential.
Until we find out how MCU plans to tie up T'Challa's story in BP2, we can only guess - but what if at the beginning of TFATWS, Bucky wasn't just mourning Steve, he was also mourning the loss of T'Challa?
T'Challa's relationship with Bucky is unique in that, despite starting off as truly trying-to-murder-each-other enemies:
1. He never treats or jokes about Bucky being semi-human. He shows Bucky that he believes him to be a good man by allowing young children to mingle freely and allowing Shuri to visit unguarded. In a country that's traditionally xenophobic, this could not have happened without T'Challa pushing it along.
2. His goodwill is directed towards Bucky because of what he has experienced, and not because of Bucky being an extension of Steve Rogers. He was the first person, aside from Steve, to see the value of Bucky Barnes as a man in his own right, rather than Captain America's best friend.
3. He and Steve are the only 2 major characters who believe Bucky needs healing and not redemption. In fact, T'Challa believed it so sincerely that he enlisted 2 of his most trusted people - his sister and a prominent member of his own guard - to help.
4. He makes the effort to understand Bucky's pain and sympathises with his wishes. Even though he says "the White Wolf has rested long enough," both his and Okoye's expressions were pained when Bucky opened the box. T'Challa understood Bucky's reluctance to go back to battle, and felt strongly enough about this that he, as king, came personally to ask Bucky to join the fight.
5. He allows Bucky to have open choices. The fact that a prosthesis was readily available, and that Bucky's first response to it was a grim "Where's the fight?" suggests that Bucky was closely involved in the decision to not use the prosthesis for most of his time in Wakanda. Instead of pushing the recommendation, T'Challa respected Bucky's boundaries and only revisited the issue when the choice was no longer feasible.
6. He allows Bucky liberties with formal addresses. Even Steve and Everett Ross greet T'Challa with "your highness", but in Bucky's exchanges with both Shuri and T'Challa, he does not address them by their title, nor does he salute them as other Wakandans do their royalty. Bucky even offers for Shuri to call him "Bucky" instead of "Sergeant Barnes", which is unusual unless the boundary is first broached by the higher ranking person. The over-familiarity is uncharacteristic of Bucky and is more likely to be part of T'Challa's efforts to treat him as a friend rather than let hierarchy hang between them.
7. He rewards Bucky's efforts to learn their language and culture with the moniker of "White Wolf". This is an endorsement that signals to other Wakandans that this foreigner has been welcomed into their community and should be respected.
8. Bucky reciprocates T'Challa's kindness with unprecedented faith and respect. When Raynor reprimanded Bucky for not trusting people are there to help him, she's wrong - because for the last 2 years he was able to settle in a completely foreign country, allow himself to be vulnerable before them, and accept the help they render. When T'Challa gives him the arm, Bucky trusts T'Challa enough that he doesn't even question why or who they are fighting.
T'Challa is the kind of amazing soul who treats both his friends and enemies with dignity. He took the time to understand Bucky and gave him more compassion than all the other scriptwriters characters. Bucky reciprocated this with a level of trust and deference that he rarely gives others. It's a friendship that gets overlooked because of how brief the scenes were and will probably be ruined by another MCU retcon down the track.
We don't know what's happening within Wakandan borders right now, but Ayo warning Bucky to stay away suggests that T'Challa either can't or isn't around to defend him any more. So it is quite possible that Bucky was in such a bad state mentally at the beginning of TFATWS because he's not only lost Steve, he's also lost one of the most generous friends he's made since coming out of Hydra - either physically or figuratively.
Still salty about TFATWS making Bucky break Zemo out of jail for no justifiable personal reason.
