How long does it take to dissolve the epoxy?
DMF (1,1,-dimethylformamide) wouldn't actually dissolve the epoxy, as an epoxy is crosslinked and is essentially one huge molecule, but it would soften it enough to let you scrape it out. I have no idea how long this would take, but might guess anywhere from 20 min to an hour, depending on the thickness of the epoxy.
How would I apply (it to) the epoxy?
I would tend to use a wooden toothpick dipped in DMF and then apply it to the individual spots of epoxy. You might also use a small syringe with a fine needle. It might be a bit critical to use what the Americans call "Q-tips" (what you illustrated) because the DMF would soften or dissolve the plastic of the stick.
I cannot find it in UK / Europe, is there a commercial name or do you know somewhere I can buy some?
I wrote you a PM on this - basically, I'd ask a local pharmacist or try the lab supply houses in the UK. This is a common solvent and the above name in brackets is the chemical name. I'm not aware of any commercial name, although there may be one or more. At any rate, even a technical grade would be adequate for your purposes, but you probably can't buy such material in small amounts (Do you have a local chemical company with a friend in the lab? That would be ideal).
Could you give any other general ideas on how to use it etc.
I'd use rubber gloves and try to avoid getting it on my hands (it can penetrate the skin), but other than that there are no particular problems. The safety angle is emphasized nowadays, but I've worked with this many times in the lab and it's no worse than many other things. The main thing to remember is that it's a very strong solvent
and can dissolve or soften nearly anything made of some kind of plastic. It's water miscible, so when you're finished you can remove it with a water-drenched "Q-tip" or the like. It also doesn't conduct electricity, so there's no danger of short circuits, although I'd try to remove it as completely as possible just because of the possible softening of the motherboard (which is a phenolic resin-impregnated paper material).
One important P.S.: I take no responsibility if someone washes his hands with this stuff or drinks it. When property used, DMF is one of the less dangerous chemicals, but everything can be misused.