For a "straightforward" adoption I'd expect
Week 1 - initial contact by adopter via website or phonecall
Followed by response by kennelling co-ordinator or phone volunteer to have a chat, make initial contact.
Week 2 - regional rep or other homechecker contacts to make appt for homecheck - subject to adopter being available - this can be done inside a week
Week 2/3 - homecheck takes place. If straightforward (fence OK and so on) homechecker will discuss adopter in detail with kennelling co-ordinator to suggest best potential match among existing hounds or whether to wait for more suitable. Sometimes at this stage rescues will also contact each other if there are specific requirements, e.g. colour and sex.
Week 3/4 If adoption requirements are straightforward, and hound is available, arrange initial meeting, trial walk or meeting with existing pets/family members. May be at kennels or at foster home
Week 3/4 If all goes smoothly, dog will be rehomed to new family with follow up calls during first week post adoption plus documented home visit one month later to assess progress. Ongoing support offered plus invitations to social events, greyhound walks, etc etc. New owners accumulate collection of collars and remember fondly when they used to be able to sit on their own furniture.
The whole process as outlined above can take as little as two weeks from start to finish. More typically, particularly when people may want to fit in appointments only on weekends, it can take a little longer. Other delays might be if the dog is recuperating from medical situation before homing, foster home availability is limited, or adopter needs work to garden before proceeding.