The OP appears to be confusing the concept of getting an appointment and the entity to whom that appointment is charged. Let me break down your question and try to answer it piece by piece.
If only one of the recipients of a nomination has an LOA, does that candidate automatically get the congressional or senatorial appointment?
I assume you're referring to the MOC slate. If one person on the slate submitted by the MOC has an LOA, that person will receive an appointment. That appointment may or may not be charged to that MOC. Thus, it is possible for others on the slate to receive appointments in addition to the LOA recipient.
And let's say for clarity that the LOA recipient has only been successful in getting a nom from one source.
An LOA recipient only needs a nom from one source. The LOA contingency is receiving "a" nomination. The source is irrelevant. As stated above, once the LOA recipient has a nom, he/she is guaranteed an appointment (if CFA/medical ok). That candidate may well be charged against that MOC (especially if the MOC gives the LOA recipient a principal nom) but it need not be the case. The LOA candidate could be charged to another source.
I've read that when more than one candidate on a panel has an LOA, they'll find nomination sources for the various candidates (such as from the NWL)
I don't really understand the question. If you're asking what happens if the MOC nominates two candidates with LOAs, the answer is that both will get appointments. One may be charged to the MOC. The other will be charged to another source. Both might be charged to another source. They don't go into the NWL/National Pool -- that group is made up non-LOA folks who secured a nom but didn't win their slate. An LOA recipient has a nom, so has an appointment, so doesn't go to the National Pool; rather, USNA must find a source to which to charge that candidate. This, BTW, is why USNA limits the number of LOAs because giving out too many wreaks havoc with the appointment process.
If you're asking what happens if the LOA candidate doesn't secure any nom from any source on his/her own, the likely outcome today is a turndown. There can be rare exceptions if USNA really, really wants that candidate. But again, the candidate won't go into the NWL/National Pool.
but my question is whether the LOA means the recipient will automatically get the appointment from a nom panel if s/he is the only one with an LOA
If you're asking whether the MOC must give a nomination to an LOA recipient, the answer is definitely no. They usually do b/c: (1) the candidate is probably very well qualified and (2) the MOC knows that if he/she gives a nom to that candidate, that candidate will be appointed. And MOCs want as many folks as possible from their districts to receive appointments -- well, most MOCs do.
If you're asking whether the LOA recipient will automatically be charged to that MOC . . . see the answers above.