Actually, gun ownership in the UK was relatively common until 1920.
The right to bear arms was actually guaranteed in the 1689 Bill of Rights:
"That the subjects which are Protestants may have arms for their defence suitable to their conditions and as allowed by law;
In 1870, the Gun Licence Act introduced the requirement that a licence was needed by anyone who wanted to carry a gun outside their home:
But there were still no restrictions on anyone keeping a firearm indoors. Incidentally, this act remained in force until 1967.
However, the government openly acknowledged that the 1870 Act was
"... chiefly an Excise matter"
So the aim was rather to raise revenue than to control access to firearms.
In the book Guns and Violence: The English Experience by Joyce Lee Malcolm presents some statistics estimating gun ownership from 1871 to 1964 in the appendix. The figures are based on licences issued under the 1870 Act. It appears that there were about a quarter of a million firearms in private ownership in 1909:
(It's also worth mentioning that it was widely believed that many people simply ignored the requirement for a licence, so the actual number might well have been much higher)
The 1903 Pistols Act denied gun ownership to anyone who was "drunken or insane", and also required a licence for firearms with a barrel shorter than nine inches.
And that was it as far as gun control went, until the Firearms Act 1920 was introduced.
So yes, it was perfectly reasonable for an unarmed Police officer to borrow a pistol from a bystander in 1909 during the Tottenham Outrage.