You should certainly have been feeling some improvement after 10 days if the bacteria you have cultured were sensitive to the antibiotic given. If not then GP should have changed them. The fact that you have the foul taste suggests to me you still have an infection. eg. If I grow strep p then it is usually sensitive to amoxil but if I grow haemophilus then doxycycline or amoxil is required. Both of these bacteria have a foul taste.
Why is GP giving you steroids in your rescue pack for bronchiectasis unless you have asthma or copd too? Steroids are often incorporated in an inhaler but we do not, in general, use them as a 'go to' drug, although we do sometimes need courses. I hope you have a good respiratory consultant with a special interest in cf/bronchiectasis, if not you should ask to be referred. Too many GPs treat us as if we have copd.
Have you got a walk in centre you could get to? If your temp goes up very high, then I feel you should go to a and e. Not ideal, I know but your infection needs sorting out.
Good luck. Please let us know how you get on.