This appears to have been a first stage appeal logged into Appeal Service's system on 14 August 2023, i.e that should be treated as the date on which the appeal was received. By virtue of paragraph 6 in Schedule 2 to the Railways (Penalty Fares) Regulations 2018, Appeal Services were required to determine whether or not to allow the appeal within 21 days from 14 August. Evidently Appeal Services have failed to meet that deadline. The consequence is set out in Regulation 16(11) of those Regulations:
“Where the relevant Appeal Panel—
(a) notifies the operator that an appeal has been allowed; or
(b) fails to adhere to the time period specified in paragraph 6 of Schedule 2, the appellant is not liable to pay the penalty fare in question.”
So you are no longer liable to pay the penalty fare. However, as a consequence of changes made in 2022 to the 2018 Regulations, you remain liable to pay “the full single fare applicable” to any travel that was authorised by the penalty fare issued to you. Whether that results in a satisfactory or anomalous/unjust outcome to your appeal depends on the grounds on which you appealed.
The fact that Appeals Service may be struggling to deal with its caseload isn't relevant. If the appeal adjudicator fails to make a decision within the time allowed, the obligation to pay the £100 penalty element (or £50 if prompt payment is made) simply falls away. Keep safe the summary of the appeal's progress that Appeal Service has supplied as evidence that Regulation 16(11) has kicked in.