My experience was after about a week the judge sent me a letter asking for additional clarification on my reasoning for changing my name as being transgender only explained a change in first name and I changed first middle and last. So I submit my thought process on why I wanted it all changed about about 3 weeks after I get the court order for a name change in the mail. This was with Spotsylvania county.
There is now a form for a change of gender for the BC, but I believe you still need a court order, Srs is two months off for me so I will have to figure out that process soon lol, but my understanding is you can mail the Richmond circuit court and initiate the process remotely which is nice as I live on the west coast now.
The SSA was a breeze to change names/gender with and the form they print out and the DMV form you can change gender marker/name on the license which took about a week and a half for me, it was quite quick.