Can trees be taken down if birds are currently nesting in them.

No, nothing should be done to disturb nesting birds but if safety is involved it may be deemed an exception, I don't know about that

No. Birds' nests are protected.

I can agree with that
A few years ago I had a broken down car in a council owned residents car park waiting to be collected by the breakers.
Before they a arrived a blackbird decided to nest under the bonnet so a notice was put on the car and it had to stay put until the young had flown even though it was not taxed, or insured anymore

You're kidding.  Boy, I wish we had some laws like that.  Here that would only apply if the bird was already a protected species.  Even then if they are not actively nesting the trees which home a protected species at only certain times of year might be cut down.

All wild birds' nests are protected; but if they're not nesting then the trees can be pruned or felled as long as there isn't a tree protection order on them, or if they're in a conservation area.