From historical evidence, it seems that the Brown rat (Rattus norvegicus) arrived in Britain aboard ships from Russia c.1728-1729. However, the archaeological evidence is poor, primarily as a result of the difficulty in differentiating brown and black rat (Rattus rattus) on the basis of post-cranial bones. Most archaeological rat bones from the 16th century onwards are not identified to species, and the authors of this paper (Dobney and Harwood, 1999: 378) suspect that some earlier finds are identified as black rat on the assumption that the date is too early for brown rat.
Evidence for the absence of brown rat from early post-medieval Britain includes the apparent larger size of black rats in the past; some appear to be comparable in size to brown rats. This may reflect the lack of inter-species competition during these periods. Only three records of firmly identified brown rat exist in the North of England. The earliest comes from the Church Street sewer excavation in York (dated to between the 6th and 7th centuries AD). However, the find was recovered from the topmost, heavily disturbed layer and may therefore be intrusive. Medieval deposits at Jarrow also apparently contain brown rat, however the potential that this is also intrusive cannot be ruled out due to the little contextual information that exists. Finally, brown rat has been identified at Skeldergate, York, in deposits dated to the early 15th century AD, but the early date of this find is not discussed in the text and so it is difficult to establish the criteria for identification or whether there might be grounds for questioning its stratigraphic provenance.
Reference: Dobney, K and Harwood, J. 1999. Here to stay? Archaeological evidence for the introduction of commensal and economically important mammals to the North of England, pp 373-387 in Benecke, N (ed.). The Holocene History of the European Vertebrate Fauna. Verlag Marie Leidorf GmbH: Rahden/Westf.