I suggest that you want to start with United, given that you want to save your AA miles for future travel. Should you find availability, your UR points can then be moved into a UA account. They usually transfer with near immediacy.

You don't want to use your BA miles because BA charges a heavy surcharge that is almost as much as a coach fare.

Southwest does not have European fights.

US requires that a RT be booked. I haven't looked up their award chart, but I believe that international coach RT is more than 36K.

You want to identify UA's international routes and look for availability on them. At a start, UA flies into LHR, FRA, MUC, AMS, BRU. It's pretty easy to look at their award availability chart. You can easily check for 3 seats on their city pairs. United shows a month at a glance, so it is quite easy to scroll thru the months of interest to you.

UA charges 30K plus tax for a one way coach ticket to Europe. You can move 10K UR points into your account, giving you 30K, enough for a one way ticket for one. You move the rest into your wife's account, giving her enough for the other 5 one way tickets.