On the
AIT website FAQ it says:
"When an appeal is allowed, the Asylum and Immigration Tribunal (AIT) will forward the Tribunal determinations to you, your representative and the Respondent (the UK Border Agency ). This is so that any party who wants to ask for a review of the determination on a point of law can do so. The AIT aims to do this within 12 working days of the hearing. The UK Border Agency will let the Entry Clearance Officer know of the decision. The UK Border Agency has 5 days in which to challenge the determination. If they decide not to do so they will send a copy of the determination to the Entry Clearance Officer and
aim to do this within 4 weeks."
Also:
"When the Entry Clearance Officer (ECO) receives a copy of the decision, they will link it to your file and take one of the following actions:
They will contact you regarding the issue of your entry clearance. The visa section will aim to contact you as soon as possible but this can take up to 4 weeks.
When the Immigration Judge has allowed an appeal the ECO will confirm whether you still wish to travel, the category of entry clearance required and whether there has been any change of circumstances. This may require another brief interview and the need to see further documents. The entry clearance should then be issued unless there has been a significant and material change of circumstances since the refusal decision, or a material deception has come to light of which the Immigration Judge would be unaware."
So ideally eight weeks (plus five days for contesting the appeal decision), but going by members' experience, a lot longer.
I received my decision on 4 April so I guess I have a long wait ahead.