Assumptions: I am assuming this is in refer to US, Canada, Mexico style plug (NEMA 1-15 or Type A) on an correctly working electricity utility service (aka hydro, AC mains, etc.) using single phase 120 Vac @ 60 Hz.
being unable to insert it due to the UPS being seemingly not designed for this type of plug,
This suggests that you (unintentionally) attempted to insert a polarized plug incorrectly. A polarized plug has one blade slightly larger than the other.
Nearly all, if not all, NEMA 5 (Type B) receptacles (i.e. wall socket) are also polarized (as well as being grounded), including those found an UPS.
I plugged my laptop AC Adapter power cable into a UPS, causing sparks to blast out
It may of been an inrush current, or rapid capacitor discharge, perhaps enabled by the accidental reversal of the Line (aka "Hot" or Live) and Neutral plug connectors. Causing a unintentional low-impedance path to discharge capacitors in either the output stage of the UPS (e.g. LC output filtering)
or part of the laptop's power supply's input section.
Would this be likely to have caused component damage and/or failure within the computer itself?
Likely, no, but possible. I would expect the mostly likely place of damage is with the power supply / charger unit itself.
What means (...) might I be able to use to determine conclusively whether or not damage had occurred and what steps would need to be taken to replace or repair any damaged hardware ...
The most complete or accurate method would be to have the power supply / charger unit inspected by an electronics technician familiar with switch-mode or switching power supplies). If there is no damage to the unit, it is unlikely (far less likely) to have damaged the laptop.
The most basic test would be to use a voltage meter (typically a VOM or DMM) to check that the the charger / power supply is still producing the correct DC voltage output. The most common failures would be to produce no (zero) output voltage, or like commonly, too high an output voltage compared to it's label.
Note that most supplies are produced to approximately 5% tolerance, so a 18V output at 5% would be valid with an output in the range of ~17.1 to ~18.9V. Most laptop supplies are regulated outputs as far as I know, so they should not necessarily need a load to produce the correct voltage (common with many cheap old fashioned unregulated wall-wart AC power modules).
If the laptop's power supply catastrophically failed it may of caused voltage surges to be sent to the laptop beyond the laptop's power connector's input protection.
The most likely resolution would be to replace the charger / power supply unit with a equivalent model with valid safety certifications (E.g. Underwriters Laboratories, Canadian Standards Association, etc.) which available from the original equipment manufacturer or from third-parties retailers online.