I think your budget and the duration of your trip should also play into your choice of destination.
I've not done a lot of European birding outside of Italy, so I don't have much comparative experience, but my best non-Italian birding in Europe was in Extremadura, Spain, in spite of it being both during a very poor time of year for vocal activity (early September) and under extreme weather conditions (afternoon highs around 110 F). It would be very easy to find cheap flights from the UK to Madrid or other Spanish cities, and car rental and accomodations are inexpensive. Plus you wouldn't have to do too much driving, so if you only had 3-4 days you could still cover the area thoroughly. At least here in southern Italy, there isn't that much drop-off in vocal activity between early and late June, so I don't think it would be a major issue in Spain, either. I guess the wildcard would be the weather, as 100+ degree heat would definitely be possible in late June.
I've not been to Finland so can't speak from experience, but my impression is both that the distances involved and the costs would be quite a bit higher than Spain or Mediterranean Europe.
And just to suggest an alternative destination, flying into Rome then spending a few days in Abruzzo would certainly get you some really good Mediterranean and montane birds (things like White-backed Woodpecker, Collared Flycatcher, Rufous-tailed Rock Thrush, Rock Partridge and White-winged Snowfinch) plus a very good chance of Brown Bear and a decent shot at Wolf. Adding a day on the coast south or north of Rome would get you a fair variety of waterbirds. You'd be hitting shorebird migration at its low ebb, but most Tringas are on their way south already, so things like Spotted Redshank or Marsh Sandpiper would be realistic targets.