Pruning trees (apple and plum)

Apples can be pruned now but you need to know the variety in order to prune it the right way.
Plum - this needs to be summer pruned.

As above - apples Nov-March, or early August.
Plums early summer. Never winter.
As two different people on how/when to prune and you'll get two different answers. What works for some people doesn't work for others.

Type and timing of pruning with apples/pears depends on what you want to achieve. Some varieties respond better (or worse) to certain types of pruning.

Pruning also depends on the size and age of the tree - and why you feel the need to prune.

I suspect that you're referring to long-neglected trees that have grown large and untidy.
Such trees can need several seasons worth of pruning to make them truly attractive specimens. A single hard prune will just make them fight back and grow in an unattractive way, with lots of upright, dense, unfruitful growth.
Snipping at them with hedge trimmers is a very bad idea; it will just make a dense/bushy crown that is very shady, damp, diseased and unfruitful. Such "haircuts" often remove a lot of the fruitful wood, which can result in low or poor quality yield for a few years.

A proper prune consists of taking a walk around the tree to see what you're dealing with. Locating dead, diseased and bady placed branches.
Then work out how you can remove those dead/diseased/badly-placed branches with as few cuts as possible.
Fewer cuts give less chance of disease entering a wound, but also help to calm down the number of places where the tree will "fight back" and help to keep the tree nicely shaped.

There are several priority rules for pruning an older tree.
It is generally best to prune old trees in winter.

1.
Remove all dead twigs and brances. Cut back to healthy wood.
Burn the dead wood.

2.
Remove all badly damaged or diseased twigs and branches.
If that has taken between 1/4 and 1/3 of the living part of the tree, STOP! The tree can't take any more that season without risking problems.
Burn the diseased branches. Don't be tempted to make a logpile for Mr hedgehog because the diseases from the rotting wood will re-infect your tree - especially as your tree will have some large pruning wounds that can take a couple of years to properly heal.

3.
If you still have some of your 1/4 to 1/3 pruning allowance remaining, remove crossing, rubbing or tangled twigs and branches.

Next summer, any vigorous upright-growing shoots should be completely cut out in early August.

If the tree hasn't been pruned for several years, you'll probably take 2-3 seasons to get the tree back to normal.
If you take too much wood, you'll either kill the tree, cause it to cease fruiting for several years, cause very vigorous and unfruitful upright growth, or cause "bitter pit" in the fruit. Maybe a combination of the above.

Thank you! I have no idea what the varieties are as they came with the house and are LONG established (40 years). We had a decent crop but as I didn't learn anything last year till too late we didn't prune them. The woman who had them before was infirm for the last few years so it's been a while. I guess I'll have a look this weekend.

Is there a sure way of telling a branch is dead?

If you're not sure how to tell dead from live wood, I'll be honest (please don't be offended) and I suggest that perhaps a professional should come and attend to your tree.

Normally, it'll show ALL of the following on dead wood:

Loss of bark (but some branches can lose bark without being dead)
Peeling bark (but some can peel without being dead).
No buds is a sign of dead twigs.
Shrivelled twigs that snap cleanly instead of bend and break in a stringy kind of way.
Go find a few dead twigs that have fallen under the tree and see how they snap cleanly, with a "crack". Then snip a few twigs off the tree and see how they bend, rather than snap. Even if they snap, it's quite a quiet "click" and the bark still holds it together.
You will also note that on live twigs, the inner wood is moist with a greenish layer under the bark, then it's alive and healthy.
You can also "shave" a piece of bark off the tree and if it's alive, it will be moist and have a greenish layer underneath.

If in doubt about dead branches, why not leave the tree until next winter, but during the coming summer, you can see which branches are dead and mark them, for removal next winter.

thank you fb!!

Of course I wouldn't be offended. I've no intention of killing the tree that arrived with the house (it was one of the main reasons to keep it)

I'll have a quick look tomorrow (if not snowing too much) and maybe take some photos for here.