I'm very sorry your Tilly is sick.
Sick budgies need warmth to help conserve their energy, this is regardless if they are bleeding or not.
Sadly, there is only so much we can do to help. We are not avian vets and even if some of us were, your budgie would still have to be physically examined and this is not something that can be accomplished online.
If you want to give your Tilly the best chances for recovery, you will have to take action, do the right thing and have him examined by an avian vet specialist, so that he is properly examined, diagnosed and treated.
Having the financial means to cater to a pet's medical needs is part of the very basics of pet ownership.
It's simply not realistic to think a given pet will never need to be seen by a vet, a pet bird is not any different from a cat or dog, they will also need routine check ups and vet visits in an emergency situation.
I hope you will be able to take Tilly to an avian vet soon and I'm wishing him a steady and full recovery.
If you are unable to have these needs covered when it comes to providing full care to your pet birds, then you may need to reconsider the ownership over them.