Spag, if at any time you are not happy with the way a rescue has handled the adoption process, then don't hesitate to go somewhere else. GRWE have a rock solid reputation. RGT branches vary considerably from branch to branch - anecdotally, I hear good and hair raising. Locally, I know people in the Welsh ones.
All rescues vary slightly in how they handle things after the homecheck, but they should be keeping you informed about the process and giving you realistic timescales. Generally, I'd expect the homecheck to clarify exactly what sort of dog will suit your family circumstances, and any special requests you may have (cat friendly, must juggle/fire-eat etc) - this then gets fed back to the kennelling co-ordinator who will be aware of all the various dogs who are in either foster or kennels, and would be suitable.
They would then call you with usually a shortlist of one or two dogs and would arrange a meeting for you and the family - sometimes this is at the foster carer's home, sometimes the dog will be brought to you, and sometimes it will be in a neutral territory such as local park, or even at the kennels themselves. A meeting is then arranged, sometimes with a trial walk, and sometimes with a second visit too. The rescue will be very happy for you to take your time about this decision. If the dog is being fostered, there may well be further consultations between the foster carer and the kennelling co-ordinator too. Eventually, a match is made and the adoption is go. Of course, sometimes if there are very simple requirements, and an easy dog this can be quite a quick process, but all greyhound rescues will be used to adoptions taking a while. You should also factor in that for many the people concerned are volunteeers so this may slow things down a bit, but even so they still should keep you abreast of things.
I'm sorry things didn't work out with Robbie, but I'm sure Operation Grab a Grund is only postponed not cancelled. 