Hi there, you are through to Dr. Jake. Im a UK based vet. Im sorry to hear of your concern with your budgie, Titch.
Budgies are very good at hiding when they are ill and when they tail bob and have a wheeze this can be a sign of concern, most likely a respiratory infection
there could be a few things going on though this would be hard to determine without a full examination.
1) a common cause of your budgies symptoms is an infection- we see these commonly in small birds such as budgies and the most common is a respiratory infection, they are usually fluffed up and their tail feathers move up and down as they breathe- this is due to them needing to put more effort in to breathe due to the infection.
A common cause of this can be avian chlamydia though many bacteria can cause this.
This can be prevented in budgies a few ways: diet is important - a lack of vitamin A makes the lung tissue weakened and more prone to infection. make sure you provide an adequate diet, fresh fruit and veg and a high quality multivitamin and mineral supplement in the water such as Avipro avian. A seed only diet isnt adequate for budgies and generally makes them become sick. Seeded diets can be dusty also affecting the respiratory system.
Regular cleaning of the cage to prevent fungal spore build up and to keep ammonia levels low will help - these are toxic to the respiratory system and can lead to respiratory disease if levels are high. Many bird owners will have an air purifier in the room their birds are kept in to help with air quality as they are very sensitive to air pollution in the room- if you get one of these make sure its an animal safe one.
2) chronic malnutrition can cause these signs, apart form vitamin A vitamin d3 and calcium are important - this also can be kept at a good level through adequate diet and also water supplementation.
As titch is already showing signs of concern i would recommend booking an appointment with a vet today who has experience in birds, they can assess his breathing and prescribe appropriate medications which will likely be antibiotics and anti inflammatories, they may also recommend nebulisation with F10 solution at a concentration if 1ml f10 to 250ml water x3 daily for 10 minutes at a time - this will need to be done in a small container.
i hope all this information helps