FirstTimeAdopters I am sorry you are having this problem and can imagine it is very stressful indeed.
However, I would be very careful about complaints or publicly naming any agency. I am sure in your frustration it may seem like a good idea but in reality I think it is a very bad idea.
Can I ask, when you say 'Second match that had broken' do you mean that you got to matching panel, were approved and then the match broke down? Or do you mean that a child was mentioned, you expressed interest in the child and then things did not proceed, or they started to proceed and then stopped. And the exact same thing happened for a second time?
If you are not happy revealing this level of detail here, please do not reveal it here.
I only mention it because they are very different things. I am trying to gauge if your frustration is just the normal frustrations with adoption or a totally high level of problems in you agency.
Re "...continuous delays" you mention "...change of staff, broken wires, miscommunication". I think change of staff is quite common in social services and people going off on sick leave seems to be quite common too.
Yes, it is I am sure devastating to be through a process that keeps you waiting and doesn't deal with you in a timely manner, but sadly I think this can be the case with adoption. When you say "We have invested months in this and are now back where we started" Do you mean months pursing these two children, one after the other?
I think if you want to change agency you should explore this quietly, the new agency will have to be in contact with your current agency.
If you have been approved for several months can you look for children from any area in England or Wales? We adopted about two and a quarter years ago and that was the situation when we adopted. So we could look for details of children from any area and then, whether your agency is a country council, local authority or voluntary agency, they will be paid for connecting you to a child (from a different country if country council etc). So it is in their interests to do this.
I just wonder why they are doing such a bad job, is it incompetence or lack of staff (or is it related to the area the children come from or staff from there) or something else. Your choice of the word 'continuous' with 'delays' suggests it might not be just general incompetence.
Anyway, all I will say is that whatever agency you go with they will be in touch with your current agency so I would not burn any bridges, even if you know you are in the right, you need to show you can work with them or that you can move on and work with another agency without being bitter and causing issues. I am sure this 'advice' will not be popular but for the sake of your sanity and your adoption aspirations I would see if you can do this.
This is unless you feel that your are being victimized for some specific reason and you can talk with your agency about your concerns.
Whatever you do I would be very cautious and get their official take on what has happened in your specific case (ideally in writing, by email etc) before talking to other agencies.
I really hope it works out well.