If statement not working

The & means concatenation as opposed a logical AND

Also, unlike Excel, you do not need to nest your IFs.

also unlike excel you can use spaces  and new lines to make the formula more readable, 

IF(

IsNull([Related Order]) AND [Type]="Non-conformance Report (NCR)",

  "NCR-" & [Tag] & "-" & IF(IsNull([Snapshot]),[Snapshot Override],[Snapshot]),

IsNull([Related Order]) & [Type]="Material Problem Report (MPR)",

  "MPR-" & [Tag] &"-" & IF(IsNull([Snapshot]),[Snapshot Override],[Snapshot]),

not IsNull([Related Order]) & [Type]="Non-conformance Report (NCR)",

  "NCR-" & [Plant Number] & "-" & IF(IsNull([Snapshot]),[Snapshot Override],[Snapshot]),

not IsNull([Related Order]) & [Type]="Material Problem Report (MPR)",

  "MPR-" & [Plant Number] & "-" & IF(IsNull([Snapshot]),[Snapshot Override],[Snapshot])

)