To be eligible for the FHS a person must (1) have mortgage approval with a Participating Lender, and (2) avail of the maximum mortgage available from one of the Participating Lenders (within the 4x gross income limit set by the Central Bank's macro-prudential rules).
The LAHL is targeted at those who cannot get sufficient funding from regulated financial providers to purchase or build a home. These include those Participating lenders of the First Home Scheme. Essentially, to obtain the LAHL, a purchaser must demonstrate that they are unable to obtain mortgage approval or a sufficient mortgage from a regulated financial provider.
Therefore, a person cannot be eligible for both the FHS and LAHL.
I can confirm, however, that the LAHL can be used to buy homes which are made available under the Local Authority Affordable Purchase Scheme. It should also be noted that a first-time buyer of a new home under any State-supported scheme could also be entitled to further support from the Help to Buy tax rebate operated by Revenue.