I don't think this is generally solvable but maybe it is so here goes.
The original situation was this:
SELECT something, big_field, complex_function(big_field), rank FROM t1
UNION ALL SELECT something, big_field, complex_function(big_field), rank
from t2 ORDER BY rank LIMIT small_number;
This query first fetches all big_field datums and does all
complex_function() calculations on them, then orders then by rank, even
though I actually need only small_number of records. There are two
problems here: first, selecting for all big_field values requires a lot
of memory, which is unacceptable, and then, running complex_function()
on all of them takes too long.
I did get rid of unnecessary complex_function() calculations by nesting
SELECT something, big_field, complex_function(big_field), rank FROM
(SELECT original_query_without_complex_function_but_with_big_field ORDER
BY rank LIMIT small_number);
but this still leaves gathering all the big_field datum from the
original query. I cannot pull big_field out from this subquery because
it comes from UNION of tables.
(I cannot limit big_field with substring() to reduce memory usage
because it messes up complex_function()).