Steve,
Life sure is complicated, isn't it? Don't worry, it will all work out in the end so stay confident on that front.
I have been in Finland for over 3 years now. When I first moved here, finding jobs without Finnish was more of a problem. But if you're educated and you have a strong work history (as I do not), then finding a job knowing only English is not as daunting as you might think (although having at least one other European language is usually preferred).
Check out these links for jobs:
A lot of them are listed in Finnish, but it's not indicative of the working language. Working language in most academic and business jobs is English. When you write your motivation letters for job opportunities (as you find them) make sure to emphasize in some way your native English speaking skills. That is a huge asset here. Finland has such a relatively small population that in order to expand they do a lot of international business, which requires, guess what- English! That's how I got my first job here at the Forest Research Institute (that and I'm studying forestry).
If you aren't certain, use google translate, highlight the whole page, and see what opps exist. It's heinous and tedious to go that route at first. I am guessing the baby-mamma is not eager to help the search? When you get to Finland there are several free or cheap adult language courses that will help, and it's a good place to make friends and network a bit. (see: ) Needless to say, as with all of these moves, it's easier to find a job once you're already here.
And to slightly disagree with you on the matter of where the baby should be raised/born- I am not an expert on the situation with the health care and maternity in the UK, but I can tell you that the socialized welfare in Finland will more than adequately assist your situation. Check out the Kela page in English about family benefits
Yes, I know, in many countries, US included, accepting social welfare is stigmatized, but in Finland it's the norm, so suck it up, do some research. There is probably something you're entitled to as the daddy. You can also call or email them, mostly they do speak English. I can't remember if there's a special service in English on the calling menu, but if not you can ask from the representative whence you get them on the line.
Also check the website in English. There are a lot of helpful FAQ and other topics listed there for immigrants from the EU and other similar topics.
Best of luck and happy fatherhood!
Megan