How to select posts with specific tags/categories in WordPress

I misunderstood you. I thought you wanted Nuclear or Deals. The below should give you only Nuclear and Deals.

select p.*
from wp_posts p, wp_terms t, wp_term_taxonomy tt, wp_term_relationship tr,
wp_terms t2, wp_term_taxonomy tt2, wp_term_relationship tr2
wp_terms t2, wp_term_taxonomy tt2, wp_term_relationship tr2

where p.id = tr.object_id and t.term_id = tt.term_id and tr.term_taxonomy_id = tt.term_taxonomy_id

and p.id = tr2.object_id and t2.term_id = tt2.term_id and tr2.term_taxonomy_id = tt2.term_taxonomy_id

and p.id = tr3.object_id and t3.term_id = tt3.term_id and tr3.term_taxonomy_id = tt3.term_taxonomy_id

and (tt.taxonomy = 'category' and tt.term_id = t.term_id and t.name = 'Category1')
and (tt2.taxonomy = 'post_tag' and tt2.term_id = t2.term_id and t2.name = 'Nuclear')
and (tt3.taxonomy = 'post_tag' and tt3.term_id = t3.term_id and t3.name = 'Deals')

Try this:

select p.*
from wp_posts p, 
wp_terms t, wp_term_taxonomy tt, wp_term_relationship tr
wp_terms t2, wp_term_taxonomy tt2, wp_term_relationship tr2

where p.id = tr.object_id
and t.term_id = tt.term_id
and tr.term_taxonomy_id = tt.term_taxonomy_id

and p.id = tr2.object_id
and t2.term_id = tt2.term_id
and tr2.term_taxonomy_id = tt2.term_taxonomy_id

and (tt.taxonomy = 'category' and tt.term_id = t.term_id and t.name = 'Category1')
and (tt2.taxonomy = 'post_tag' and tt2.term_id = t2.term_id and t2.name in ('Nuclear', 'Deals'))

Essentially I'm employing 2 copies of the pertinent child tables - terms, term_taxonomy, and term_relationship. One copy applies the 'Category1' restriction, the other the 'Nuclear' or 'Deals' restriction.

BTW, what kind of project is this with posts all about nuclear deals? You trying to get us on some government list? ;)

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" 
                                           )
                            )
            )