I had a lot of gouldian losses when I kept my pairs outdoors, too, and they didn't have exposure to wild birds (they were on a screened in patio). The biggest culprits in the case of my outdoor birds were coccidia and pseudomonas.
Your birds additionally have the risks assosciated with coming into contact with wild birds and insects which is likely where they got the tapeworms. The tapeworm segments that are shed in feces sort of look like small, flat grains of rice:
That obviously isn't a picture of bird feces with tapeworm, but it gives you an idea of the appearance of the tapeworm segments.
Tapeworm requires an intermediate host (insect) before it can become infectious to birds. This means that your birds did not get tapeworm through contact with wild birds directly, but indirectly, since the wild birds shed the tapeworm segments in their feces, then an insect picks up the tapeworm eggs from inside the segment, the eggs develop within the insect, then the insect is eaten by the finch and the finch becomes infected with the worm.
Intermediate hosts can include fleas (in the case of dogs and cats especially), house flies, beetles, ants, some species of slugs and snails, and mealworms. If you are feeding mealworms, it is possible your culture is infected and may need to be discarded--if you are raising mealworms, please give me the details of your set up (how they are raised, where they are raised, etc.)
Praziquantel is a dewormer of choice for tapeworms. Ordinarily in the states, you would obtain this drug by taking an affected bird to the vet. This is good for several reasons (and I highly recommend it if you can): 1. the vet can take a look at the droppings and tell you if they really are tapeworms, 2. the vet can pick up on any other abnormalities or problems you may have missed (i.e. coccidia, bacterial infection, etc.), 3. the vet can prescribe the appropriate drugs in the appropriate doses for YOUR birds and talk to you at length about ammending these problems and preventing their future occurance.
Sometimes getting to a vet isn't very feasible, however... sometimes an avian vet is not present in your area, and sometimes cost is a concern. Luckily, praziquantel is available in the form of "Worm Out gel" (a vetafarm product) online at such stores as
http://www.birds2grow.com]
If you go the route of shopping for the products online, pick up Coccivet as well (for coccidia), Ronivet-S for giardia, SCATT for mites, and avian insect liquidator (a somewhat bird safe insecticide--just don't use it around fish).
Your goal now will be to treat your birds not only for worms but for other pathogens which could be causing your losses, and then to take steps to prevent future problems.
Treating with Worm-out will help to take care of the worms. Follow the instructions on the label.
Treating with SCATT with rid your birds of external parasites. Whether or not you have seen signs of mites on your birds, it is a good idea to regularly treat outdoor birds which are exposed to wild birds.
Treataing with Coccivet will help to eradicate any coccidia. Coccidia typically causes diarrhea with marked weight loss and sudden death. It is worth treating regularly for this pathogen in outdoor birds.
Ronivet-S will help treat for some protozoa (e.g. giardia) that wild birds can transmit to your finches. Again, it is worth treating outdoor birds with this kind of exposure regularly (follow the label).
Avian insect liquidator is one way you can help prevent reoccurance of tapeworms, because it will kill many of the intermediate hosts. Another step I highly recommend you taking if possible, is to place screen around your enclosure to prevent entry of most insects. Placing a roof over a majority of the enclosure will help to limit wild bird droppings from coming into contact with your birds. Trim back any bushes, plants, or tree branches that may be encroaching on your finch enclosure(s).
Your goal is to prevent wild bird feces from entering your enclosure, and to limit access of insects (roaches, beetles, ants, flies, mosquitoes) to your enclosure as much as possible.
http://www.finchinfo.com/housing/combat ... dators.php has more tips, as well.
If I were you, I would do the following things in the following order:
1. Capture each bird and treat it with SCATT, then do a mini physical exam on each bird while you have it caught: check the feathers around the nostrils for matting/wetness/staining. Any of these findings indicates illness, usually a respiratory infection. Feel the breast bone of the bird. If the bone is incredibly prominant, this indicates severe weight loss. If the bird feels full around the bone and the bone can be felt slightly but there is muscle on either side of it, this indicates a healthy weight. Check the vent of the bird for feces/wetness/staining. These indicate diarrhea, usually. Any of these abnormal signs indicates a sick bird that you should consider bringing to the vet.
2. The day you treat for mites, also start treating all of your birds for tapeworm with praziquantel. Thoroughly clean the cage(s) at this time if it is not too disruptive to the breeding birds, and spray the cages with avian insect liquidator. If you cannot clean the cages yet, at least clean the bottoms of the cages to remove the current droppings, and spray the cages and the area surrounding with avian insect liquidator.
3. As soon as the praziquantel treatment is done, treat all of your birds for coccidia.
4. As soon as the coccidia treatment is done, treat with Ronivet-S.
5. As soon as you can do so, try to screen in the enclosures and place a roof overhead. That, or consider keeping your birds indoors where you will have much greater control of preventing these kinds of disease outbreaks.
I am pretty sure all of these treatments are safe to do in breeding birds, but double check all labels before use to be sure.
Good luck, and if you are able to take a good picture of your finches' droppings with tapeworm segements in them, I'd be most appreciative!