It was said that the cells are undergoing testing and deliveries could start as early as in the second quarter of 2022, but now it seems that it's not the case. Specifically, it's not the case for the MIC Model Y.
In China, Tesla currently uses LFP (lithium iron phosphate) batteries supplied by CATL in parallel to NCM 811 and soon NCMA cells from LG Chem's LG Energy Solution. In the initial period, the cells were supplied also by Panasonic.
The wide list of battery cell suppliers means that Tesla is very pragmatic and another supply deal with BYD would sound reasonable.
A decade ago Tesla's Elon Musk was not too worried about BYD's electric cars (at the time the company was considering expansion to the U.S.), but over the years the company significantly improved its products and remains one of the largest plug-in car manufacturers.
Who knows, maybe at some point in the future the two will become partners on some EV projects. A lot depends on Tesla's progress on its own cell formats (cylindrical 4680) and value of off the shelf solutions.
We can estimate that Tesla's battery needs in China alone already exceed 20 GWh annually, as the company produces more than 30,000 units a month for China and export to other markets including Europe.
Any significant increase in the future and especially the addition of a new higher-volume model will require basically a new battery gigafactory. We guess that Tesla already is in a process of figuring out which path to choose because the investment will take a year or two.