As of 20250224 16:32 CT, Teams V25017.203.3370.1174, I was able to copy & paste a conversation with sender and timestamps.
Select all the text (and unfortunately, other text of the window, such as the list of chats) by clicking just above the line separating the conversation from the sender(s) name(s), then move the mouse down over the conversation and use Ctrl+a which selects the entire conversation.
Then use Ctrl+c to copy it.
Paste where desired.
I find it is best to first open the chat in a new window (in the ellipsis "..." menu, upper right) as that separates the app list and chats list so those don't get copied too.
I posted the below earlier...
Cleanup: Remove all the junk before the actual part of the conversation you want. Remove any junk at the end of the parts you want.
Now I do this in Outlook with a find/replace. Your editor may have different find and replace regular expressions.
Remove all the summary lines: "ya dee da dee da ... by Firstname Lastname (and any other text)" and replace with new line...
Find:^13[^32-^126]@ by [A-Z][a-zA-Z -]@, [A-Z][a-zA-Z ()]@^13
Replace:^13
Next, append the time after the sender's name, with a separating space...
Find:^13([1-9]{1,2}):([0-9]{2}) ([AP]M)^13
Replace: \1:\2 \3^13
Note there is a space after the replace colon!
Of course, if you use 24-hour time (which I do on everything but my work computers) you'll have to modify the time find string.
In the first find, ^32-^256 are all the printable ASCII characters. Not sure what can be done with other languages using accented characters.