In medieval England, most restrictions I am aware of on weaponry are generally concerned with particular practices rather than the ownership of weaponry in general. In the 1300s, there seems to have been a great deal of concern about crime and violence in London, which was met by bans on carrying weaponry in the streets. From November 4, 1327:

The King's writ to the Mayor and Sheriffs with regard to disturbers of the peace. The King has heard that evildoers have formed unlawful confederacies in the City since the beginning of his reign, and have been engaged in beating, wounding and killing citizens and others, robbing persons of their goods, and holding to ransom those who visit the City and suburbs on business. The Mayor and Sheriffs are ordered to hold inquiries, by the oaths of good and lawful men of their bailiwicks, as to the truth of the above reports, and to commit to prison until further orders all persons indicted before them for such offences, or such as they may find perpetrating them. Dated at Nottingham, 4 Nov. Ao 1 Edw. III [1327]. (L)

And followed up on November 8 of the same year:

Writ (fn. 17) to the same forbidding the carrying of swords, clubs, cross-bows, and bows for discharging stones and clay pellets (pelotas terreas) in the City. Offenders are to be committed to prison until further orders. This prohibition does not apply to the sergeants-at-arms of the King and Queen Isabella, nor to the servants (valettis) of earls and barons of the realm, viz. each earl and baron is allowed to have his servant carrying his sword in his presence. Dated at Nottingham, 8 Nov.

Writ (fn. 18) to the same ordering them to punish such bakers, taverners, millers, cooks, poulterers, fishmongers, butchers, brewers and cornmongers as are found to be lax in their work and misteries, and to discourage by corporal punishments and other severe measures, those evildoers who go about the city by day and night with swords, clubs, bucklers, and other arms, beating and ill-treating individuals, either out of wantonness or by the instigation of others. Dated as above.

Both of these writs are from the Calendar of Plea and Memoranda Rolls of the City of London, found on this page.

So obviously there is a strong class element here, but this should not be taken as a general prohibition on regular people owning weapons. This same demographic of tradesmen would have been important for military recruitment, so many of them would be both encouraged and required to own weapons and armor. Owning weapons was not the problem. Running around the city with a sword and buckler getting into fights with people was. We see this distinction clearly in other cases:

Adam Grymmesby was committed to prison for not warning his lodger to leave his knife indoors (fn. 16), for which neglect the knife was confiscated, and when the said Adam was asked to redeem it, he refused, and showed contempt for the Mayor's summons to appear, saying he would come next day.

Proclamation for keeping the King's Peace in the City: The carrying of arms in the City except by the officers of the King and the City is forbidden, and innkeepers shall warn their guests of this on their arrival. Any one raising a disturbance close to the doors and windows of houses and shops or resisting arrest shall be taken before the Mayor, Aldermen and Sheriffs and have judgment of life and limb. No one shall be a maintainer of persons of evil character or conspirators, under pain of forfeiture. Keepers of taverns, to which evil characters usually resort, shall not keep their doors open after curfew sounded at St Martin-le-Grand, under pain of imprisonment. Taverners shall not mix bad wine with good etc. ut supra.

Travelers were not restricted from carrying arms for self-defense, but once they had arrived at their lodging, they were supposed to put it away until they left the city. There would be quite a few people with weapons hanging around inns: in times of war, soldiers from other parts of England would often drift down to London in order to find a company or retinue to contract with before departure for France (or wherever). They would therefore require temporary lodging. Unfortunately, the precise details of the military network/market that operated in London are unknown to us. The royal clerks were unconcerned with the petty details of how soldiers arrived in the army so long as there were plenty of men available to uphold the King's claims in France.