Not bad...
Zurich - AA: 30k + $78. UA: 30k + $64
Copenhagen - AA: $69. UA: $49

I can't seem to make the connections work though...
Spent the whole day on UA.com, BA.com, AA.com, Expertflyer.com
And nothing.

if you have chase ur points you can transfer them to southwest for a cheap connection to NYC or chicago or boston, to connect with your trans atlantic flight. or you can book low point trips on american inside the us with avios, very limited availability though.

No I meant, I couldn't seem to work the connections for BHX to Zurich or Copenhagen.

BA doesn't fly from Birmingham so I'd be using UA.
Even still, BA from LHR or UA from BHX, nothing would get me into Zurich or Copenhagen in time for the flight.
Only way I could do it would be to come the night before.

Then I had two additional problems- Not knowing in advance if my Schengen visa would be multi-entry or not?
And finally trying to get back to Hartford from where ever I landed.

I finally just gave up and took the easy way out.
Found a LHR-EWR-BDL flight on UA, and just picked that. I'm never in a good mood when saying goodbye and all that running around wouldn't help.
This way, I get straight home without worrying about anything else.

Sure it's $187, but the way I rationalized it is by thinking, I'm getting to U.K for 14k only. So overall, it averages out pretty well.

But you can often get around that restriction by addinig a small flight from your nearest neighbor to home. Shorts flights like YVR-SEA can be really handy to get around the return to originating country rule in the ANA program.

In a little more detail: ANA allows tickets of the form www-xxx, yyy-zzz as long as www and zzz are in the same country (and perhaps yyy-zzz is crossing a border). If you want a one-way, then book a nice long www-xxx segment and the shortest/cheapest yyy-zzz segment you can find.

will be a perfect time. You can catch the BPL live. don't miss.