Had this very case a couple of years back on an A320 with the old LITTON ADIRUS (no GPS installed).
ADIRS #3 inop as per M.E.L. and all characteristic speeds were WAY too low (10-20 knots). Unfortunately Vls, Vaprot affected as well. An alpha-floor scenario would have been very interesting this day
Characteristic speeds are calculated by the FACs taking into account the FAC-Grossweights (these can be looked up on the newer ships using the GWFK/GWFL AIDS Alpha Call Up).
The FAC-Crossweight is calculated using the CPT AOA, FO AOA and STBY AOA-Values. Both FACs receive all 3 AOAs and do some biasing. The AOAs are accesible even on the oldest ships using the ALPHA CALL UPs AOA/AOA3.
As the AOA meassurement is prone to huge errors (an AOA-vane 'off' by 0.1 degrees results in a GW error of +/- 1 ton) the GW is just as likely to be affected.
If one out of three AOAs is off the mean value, this can be covered by the FACs by blending out this AOA-value.
If one ADIRS in INOP however it's Crew Operating Procedure to set the IRS Mode Selector to OFF. Unfortunately on the A320 you loose the ADR part when selecting the ADIRU to OFF, there is no way to isolate the IR part alone (as opposed to the A330/340 where that is possible).
In our case it was found out, that the FO AOA was offset by about 1.2 degrees which was never caught while all the ADRs where working. As ADR was OFF now the STBY AOA was no longer taken into account and the FACs derived the GW as an approximate interpolation between AOA1+2.
During Departure we both thought 'Man that green dot speed is low', but only when we saw the MCDU Message 'Check GW' we started to really see the problem. During Take-Off you can check plausibilty of the FAC-Speeds by checking Vls which should be in the vicinity of 100+(GW-30), ie. TOW 65 to -> Vls ~ 135 kts.
As a result of this incident the M.E.L. now says 'Characterisic Speeds may adversly be affected - check QRH' an we have a limit of 8 Knots between any FAC-calculated and QRH speed.
BRGDS,
MAX