I can't address whether US postal is he best method for shipping to UK, or domstic, but I can advise you to check if something the size/shape/weight of a packed gun will still ship with them internationally. This I NOT whether the will handle guns but new physical size restrictions.
Lat month the US postal service dramatically overhauled it's parcel post with major restrictions on size/weight/shape and elimination of international ground service. Of course, fees have increased dramatically for what they still handle.
In any cases, they will not handle most packages much greater than 1 cubic foot and weighing somewhere around 15-20 lbs (hard to tell exactly since the new formulas are a nightmare & it is best to plug in the numbers on their website to see if they will still handle the item).
Domestic Parcels are a bit less restricted but still affected.
Some examples from my recent experience;
Prior to service revision I sent a modest size/weight package (24"x18"x18" 35 lbs) to NZ via insured Air parcel post. It cost around $60 to ship.
Last week I did an estimate for a smaller package (12"x12"x12" or 1 cubic foot, and 15lbs weight) to the same place. It would have cost around $80 via Priority mail (the "new/improved" intl air parcel service) to the same location. The larger package will not ship.
Prior to new service I shipped a package (24x18x18 and 9 lbs) to western OZ for about 35$ by surface parcel post (air service was about 65$).
After the changes, US Postal would not handle it at all since it exceeded their new size rules.
I would find out the exact size/weight/destination of your package, go to the US Postal website and plug them into the shipping calculator to see IF they would handle them at all and how much I would cost if the do.
