If a bailiff finds a key under the mat, can he enter using the key?

No

Enforcement regulations are silent on whether the bailiff can use a key found under a doormat, but case law does offer guidance on whether the entry is legal.

A bailiff cannot put his hand through a cat flap or letterbox to draw back a bar or fastening to open the door or window, because it is burglary and trespass.^[1]

The use of a landlord's key to gain entry is unlawful.^[2]

If the key is found, but is not in the lock, the entry would be unlawful.^[3]

Improper use of keys to gain entry is illegal.^[4]

If a key is in the lock, the bailiff may gain entry using the key.^[5]

^{[1] Ryan v Shilcock [1851] 7 Exch 72
[2] Miller v Curry [1893]
[3] Welch v Krakovsky [1919]
[4] Alford v Thrupp [1906] 67 EG 226
[5] Ryan v Shilcock [1851] 7 Exch 72}