Mapping values from the same table recursively

Try this:

let
    Source = Table.TransformColumnTypes(Table.FromRows(Json.Document(Binary.Decompress(Binary.FromText("i45WMlTSUQKiRKVYnWglIyALJJAE5hkDWSCRZKXYWAA=", BinaryEncoding.Base64), Compression.Deflate)), let _t = ((type nullable text) meta [Serialized.Text = true]) in type table [ID = _t, #"From ID" = _t, Type = _t]),{{"ID", Int64.Type}, {"From ID", Int64.Type}, {"Type", type text}}),
    lookup = (n, tbl) => if tbl{[ID=n]}[From ID] = null then tbl{[ID=n]}[Type] else @lookup(tbl{[ID=n]}[From ID], tbl),
    AddCol = Table.AddColumn(Source, "Original Type", each lookup([ID], Source), type text)
in
    AddCol

The key is the recursive function defined in the query:

lookup = (n, tbl) =>
    if   tbl{[ID=n]}[From ID] = null
    then tbl{[ID=n]}[Type]
    else @lookup(tbl{[ID=n]}[From ID], tbl)

Note that you need to use the @ symbol to reference a function from within the definition of that function.

See also:
https://radacad.com/fibonacci-sequence-understanding-the-power-query-recursive-function