There is mention of this on British Listed Buildings. It states the date it was built was c19th-early20th century. Coflein puts the date it was built between 1913 and 1937:
https://www.coflein.gov.uk/en/site/16604/details/penrhyn-old-hall-north-lodgeBarratt Obadiah Carrington Sellars was born in 1865 and his birth was registered in Birkenhead. In the 1901 census he is living in Moss Brook House, Audenshaw, Lancs. He is aged 35 and a widower. His occupation is Chemical Manufacturer.
In the 1911 census he has married again, to Dorothy, and has 2 children. He continues to be Manufacturing Chemicals.
Barratt and Dorothy married in Holt Trinity church in Cheltenham on 24 June 1902. He was a widower, aged 36 and she was a spinster, age 20. His profession is shown as Manufacturing Chemist.
His correct name is Carrington Sellars.
Barratt Carrington Sellars died in 1927 and the Probate details state, "Sellars Barratt Carrington of Mossbrook House Fairfield Manchester and of Penrhyn Hall Penrhynside Llanrhos Carnarvonshire died 23 February 1927 at Mossbrook House. Probate 17 May to Dorothy Georgina Sellars widow William McGuinness Steamship Company's Manager and James Thornily Fox Solicitor. Effects:£76372 11s 3d."
That suggests the North Lodge was built pre 1927.
Helig