Some Twin Peers and Baronets I found in Complete Peerage and Baronetage: (This is not meant to be a complete list. I do not mark twins in myh files, so I can find only the ones listed in Cokayne with the word "twin".
Twins who both succeeded:
William Thellusson, 2nd Lord Rendlesham, was succeeded in 1832 by his younger twin brother, William, 3rd Lord Rendlesham.
Sir Edward Dineley-Goodere, 4th Bt. (1729-1761) was succeeded by his younger twin brother, Sir John Dineley[-Goodere], 5th Bt. (1729-1808/9).
Twins of whom only one succeeded:
Anne Katharine, 7th Countess of Antrim (1778-1834), had a younger twin sister who died before Anne did
Charles (d.1695) and William (d.Jun 1740, two months before their father) Knollys, born 15 Oct 1694, respectively called Viscount Knollys as heirs apparent to the so-called 4th Earl of Banbury
Henry Conyngham, 3rd Lord Conyngham, later Earl and Marquess Conyngham, born 26 Dec 1766, had a twin brother, Hon. Sir Francis Nathaniel Burton Conyngham
William Stanhope, 2nd Earl of Harrington (1719-1784), was the only survivor of twin sons, and their mother died in childbirth.
On the death of the wife of Charles Kinnaird, 5th Lord Kinnaird, "At that date she was alleged by her husband (after having been 18 years childless) to have given birth to twin sons who ( if, indeed, they ever existed) were promptly declared by him to be both dead, when an action was brought against him by the next heir for forgery." He was succeeded by a cousin.
Robert de Beaumont, said to have been created Earl of Leicester in 1107, had twin eldest sons; for some reason the English Earldom devolved on his younger twin son, Robert, 2nd Earl of Leicester, while the older twin, Waleran, succeeded to their father's French territories, and was apparently later created Earl of Worcester.
The 5th Earl of Longford had twin eldest sons born 19 Oct 1864. The elder, William, Lord Silchester, died in 1876; the younger succeeded his father as 6th Earl.
Edward John Stanley, 2nd Lord Stanley of Alderley (1802-1869), was the elder of twin brothers.
Francis, 10th Earl of Moray, had a twin brother who was ancestor of the 15th Earl.
Thomas Noel-Hill, 8th Lord Berwick, was son of a younger twin brother of the 7th Lord.
Sir John Davie, 9th Bt. (1798-1824), was one of the twin sons of the 8th Bt.
Sir Martin Stapylton, 8th Bt. (1751-1817) was the younger of twins, his elder twin brother having predeceased their father.
Sir John Reade, 6th Bt. (1762-1789) was one of the twin sons of the 5th Bt.
Sir Francis Curson, 3rd Bt., was the younger of twins, but his elder twin brother died young.
Sir John Shaw, 7th Bt., was born 1829 as the posthumous elder twin son of the brother of the 6th Bt.
Sir Rupert Ford, 5th Bt. (1877-1948), was the elder of twin brothers.
Sir Frederick F A Steele, 5th Bt. (1787-1876) had elder twin sons, born in 1810, who both went to the West Indies ca 1830 and disappeared.
Then there was the famed Douglas Case:
"Lady Jane Douglas was b. at Douglas, 17 March 1698 ; m. (as his second wife) 4 Aug. 1746, Col. John Stewart (afterwards, 1 Nov. 1759, a Baronet) and d. at Edinburgh 22 Nov. 1753. She is said to have had twin sons in her 51st year, b. 10 July 1748, of whom tho yst. , Sholto Thomas Stewart, d. at Edinburgh 14 May 1753 , in his 5th year ; while the eldest suc. to the Douglas estates... It is remarked by Sir Bernard Burke that " many doubts had existed from the time of his birth as to its genuineness, " the Duke of Douglas himself being among those who considered the children to have been "stolen or bought. " Sir Bernard adds that " when the question is now considered after the lapse of a century, apart from personal feeling and party bias, it seems impossible to reconcile the contradictory assertions connected with the strange story of Mr. Douglas' birth or to resist the strong appearance of imposture.
Sir John Stewart surprisingly recognized the twin sons as being his and his wife's sons, the survivor of whom eventually succeeded to the great Douglas estates and was created Baron Douglas.