What a gross DB structure.
Anyway, I'd do something like this (note I prefer EXISTS to joins, but you can re-write them as joins if you like; most query analyzers will collapse them to the same query plan anyway). You may have to do some additional juggling one way or another to make it work...
SELECT *
FROM wp_posts p
WHERE EXISTS( SELECT *
FROM wp_term_relationship tr
WHERE tr.object_id = p.id
AND EXISTS( SELECT *
FROM wp_term_taxonomy tt
WHERE tt.term_taxonomy_id = tr.term_taxonomy_id
AND tt.taxonomy = 'category'
AND EXISTS( SELECT *
FROM wp_terms t
WHERE t.term_id = tt.term_id
AND t.name = "Category1"
)
)
AND EXISTS( SELECT *
FROM wp_term_taxonomy tt
WHERE tt.term_taxonomy_id = tr.term_taxonomy_id
AND tt.taxonomy = 'post_tag'
AND EXISTS( SELECT *
FROM wp_terms t
WHERE t.term_id = tt.term_id
AND t.name = "Nuclear"
)
AND EXISTS( SELECT *
FROM wp_terms t
WHERE t.term_id = tt.term_id
AND t.name = "Deals"
)
)
)