I am not aware of any specific prohibition on, or regulation of, use of the terms "inc" or "incorporated" in this country.
However, it might possibly be argued that use of either of these terms in the name of an unincorporated business could be be misleading under the new legislation which replaced the Trade Descriptions Act 1968, which was brought in under the EU Unfair Commercial Practices Directive (Dir 2005/29/EC of the European Parliament and of the Council of 11 May 2005 concerning unfair business-to-consumer commercial practices in the internal market and amending Council Dir 84/450/EEC, Dirs 97/7/EC, 98/27/EC, and 2002/65/EC of the European Parliament and of the Council and Regulation (EC) No 2006/2004 of the European Parliament and of the Council).
This new legislation takes the form of two implementing statutory instruments, namely The Business Protection from Misleading Marketing Regulations and The Consumer Protection from Unfair Trading Regulations.
It is therefore possible that a complaint might result in action by a local trading standards officer, as the two statutory instruments in question are so wide and vague in scope as to potentially catch use of any business description or terminology which might perhaps be misleading.