Generally, the compound verbs in your sentence should use the same tense, indicting the same time. To oversimplify slightly, will is future tense and would is past tense. In your sentence, both uses are in the future, after you've purchased the bicycle; so both should be will:
"Not only will it allow me [in the future] to be more independent but will [also in the future] also help me with my body fat accumulations."
This is not a hard and fast rule; sometimes a sentence will indicate both action at different times. "I now own a bicycle [present tense], but I did not own one two months ago [past tense; the present tense would be "do not own"]."
It gets a little more complicated because would can also be used in the present tense in some circumstances. "I would like to buy a bicycle today, but the bicycle shop will be closed when I get off work." This is called the subjunctive mood, which is used to express a wish or desire (as in the example) or something contrary to fact ("I would have bought a bicycle, but the shop was closed.") and in a few less-common uses. None of those uses apply to your sentence, though.
A side note: I deliberately did not fix it above, but your sentence also has another problem with parallelism. Both "not only" and "but also" should be followed by the same type of grammatical structure. In your sentence, the subject (it) is after "not only," so there should also be a subject after "but also" (making a compound sentence, with a separate subject for each verb). You can also move it before "not only," making it a simple sentence with a compound verb (two verbs with the same subject). Either of the following is correct:
- "Not only will it allow me to be more independent, but it will also help me with my body fat accumulations." (Compound sentence; note that a comma is needed before but.)
- "It not only will allow me to be more independent but will also help me with my body fat accumulations." (Simple sentence with compound verb; not that there is no comma before but.)