From your description and by looking at the budgie's posture on the second to last picture, it does seem that he is feeling under the weather.
Is he able to eat well at this point, or have you also been noticing a decrease of food intake?
Unfortunately, we can't tell with accuracy what is wrong with your budgie and in order for him to have the most adequate treatment plan, he will need to be examined by an avian vet specialist.
No matter how small or inexpensive a given pet may be, when bringing home a pet a person is taking the responsibility to care for said pet and that includes providing medical attention when necessary.
This is also extremely important when breeding any species of bird and I gather from the pics that the 3 budgies are chicks that you got from breeding one of your pairs.
I also don't have access to avian vet services in my town, but that doesn't keep me from travelling whenever it's necessary.
I hope you will be able to convince your mother and take your budgie to the avian vet.
Meanwhile, you can soak some millet spray on electrolyte solution to hopefully give your budgie a little boost: Guardian Angel // Pedialyte // Pedialyte Recipe
You can also try to see if he will eat some egg food as this is highly nutritious and is a good supplement to give to sick and convalescent birds.
I hope all goes well and your budgie feels better soon.