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The British- and Irish-English noun fish finger designates a small oblong piece of filleted or minced fish coated in batter or breadcrumbs, typically fried or grilled.

This noun occurs, for example, in the following from There are no wrong answers… We all know that children like to ask endless questions but the way in which you respond to them could play a vital role in their development, by Rita deBrun, published in the Irish Independent (Dublin, County Dublin, Ireland) of Monday 28^th September 2009 [health & living section: page 9, column 3]:

What’s a parent to do when she can’t explain why centipedes have so many legs, why the moon can be seen by day, and why, if fish don’t have hands, she served fish fingers for lunch?

In American English, the more usual term is fish stick.

Now usually sold pre-prepared for cooking from frozen, fish fingers were originally homemade—as shown by the earliest occurrences of the noun fish finger that I have found; these early occurrences are as follows, in chronological order:
—Note: In early use, the noun fish finger designated various dishes:

1-: From Wings: The Official Organ of the Women’s Total Abstinence Union of September 1898 [page 119, column 2]—as quoted in the Oxford English Dictionary (online edition, June 2024):

This is how I make my ‘fish fingers’; yours were evidently too dry.

2-: From Hints for the Home, published in The Tunbridge Wells Courier (Tunbridge Wells, Kent, England) of Wednesday 27^th June 1900 [page 3, column 5]:

Tasty Fish Fingers.—A very useful breakfast dish can be prepared as follows: Flake and mince any cold cooked fish very finely, and mix it with two-thirds its bulk of cold cooked rice; season to taste with pepper, salt, and minced parsley, mixing it into cakes with the white of an egg; shape these neatly, brush with beaten egg, cover with bread-crumbs, and fry in plenty of hot fat to a delicate golden brown; drain well, and serve.

3-: From The Housewife’s Corner, published in the Farnworth Chronicle (Farnworth, Lancashire, England) of Saturday 15^th November 1913 [page 12, column 5]:

Bouillabaisse.—Put into a saucepan six small onions, sliced, half a clove of garlic, one carrot, sliced, three pounds of inexpensive white fish, boned and cut into “fingers,” two and a half pints of fish stock, with a seasoning to taste of salt, pepper, and allspice. Put the lid on the saucepan, and simmer gently for about 20 minutes, then carefully remove the “fish fingers”’ into a hot tureen, sprinkle them with chopped parsley, give the soup one boil-up, and strain it over the fish. Serve with small squares of toast.

4-: From Woman’s Ways, published in The Leicester Mail (Leicester, Leicestershire, England) of Thursday 23^rd April 1925 [page 3, column 4]:

FISH FINGERS.

Make a little short pastry—“trimmings” can be used for this. Cut into strips about three inches in length. Spread with a little pounded fish and squeeze a few drops of lemon juice on each. Season with pepper and serve.
LYNETTE.

5-: From Dainties for the Christmas Tea Table, by Elizabeth Craig, published in The Yorkshire Evening Post (Leeds, Yorkshire, England) of Monday 14^th December 1931 [page 5, column 2]:

I start tea with sandwiches made of half toasted bread, put together with flaked tunny fish, moistened mayonnaise, and thin slices of tomato […].
To make tunny fish fingers, toast white bread on one side, spread the untoasted side with butter, then with flaked tunny fish, lightly moistened with mayonnaise, and mixed with minced pimento, in the proportion of one tablespoonful of pimento to four tablespoonfuls of tunny fish. Cover with another slice of half-toasted bread, lightly buttered on the untoasted side, and cut into fingers. Sometimes I sprinkle the filling with a little mustard and cress, or shredded watercress.

6-: From French Ways with Fish, published in The Northern Whig and Belfast Post (Belfast, County Antrim, Northern Ireland) of Monday 30^th March 1936 [page 10, column 4]:

Fish Pate.—Turbot, hake, cod, salmon, or any fish which is not full of small bones may be used for this. Skin and bone the fish, and cut half of it, choosing the best and thickest pieces, into fingers. Flake the rest as finely possible and add to it its weight in breadcrumbs and butter. The breadcrumbs should be soaked in milk and drained. Make into a paste with the butter and the yolk of one egg, season with salt, pepper, and chopped parsley, and add some chopped mushrooms if liked. Butter a casserole, put in a layer of fish mixture, and on this place a layer of fish fingers sprinkled with salt, pepper, and a pinch of dried mixed herbs. Continue in layers till the casserole is nearly full. The last layer should be the fish mixture. Place some small pieces of butter on top, put the cover on the casserole, and bake in a moderate oven for two hours and a half.

7-: From Dainty Sweets, published in the Northern Weekly Gazette (Stockton-on-Tees, County Durham, England) of Saturday 27^th March 1937 [page 13, column 2]:

Fish Fingers For Invalids

Take fillets of plaice, and form each into a little roll. Stand these on end in an enamelled plate containing two tablespoons of milk. Fill each hole formed by the roll with a few crisp breadcrumbs. Put the plate over a saucepan of boiling water, and cover. Let it steam for fifteen minutes, then squeeze a few drops of tomato or lemon juice into the little hole, and insert a little pat of best butter. Re-cover and steam another five minutes, when the dish is ready to serve. A little mashed potato served with this makes a very nourishing and sustaining dish for an invalid.

8-: From This Week’s Recipes, published in The Walsall Observer and South Staffordshire Chronicle (Walsall, Staffordshire, England) of Saturday 5^th June 1937 [page 12, column 6]:

Potted Fish Fingers.
Sent by Mrs. G. Watson,
4, Station Bridge, Pelsall.

Ingredients: ½lb. fish, ½-teacupful of breadcrumbs, ¼lb. tomatoes, 4 large potatoes, 2ozs. butter, a little mustard.
Method: Steam the fish, bone it and then mash it up. Boil the potatoes. Mash them with a little butter. Put tomatoes in boiling water for a minute and take off the skins. Place all ingredients in a bowl, add pepper and salt to taste. Beat well; you may add a little mustard or any kind of sauce. Mix enough breadcrumbs until it is a smooth paste. Make toast and spread or cut into fingers or spread on bread and butter.