Autofocus does not work

No Nikon lens ever zooms automatically, so I assume you mean focus.
The D5100 lacks an autofocus motor, requiring the lens to have one. The older 70-300mm lenses lack this motor, expecting the camera body to have one.
The D5100 will only autofocus with lenses designated AF-S or AF-I. It will not autofocus with older AF-D and AF-G lenses.

The D5000 does not have a focus motor in the body, relying on the lens to have one. The old 70-300mm G lens does not have a focus motor, relying on the camera to have one. Thus, this combination will not autofocus.
The newer 70-300mm AF-S G lens, as well as the even newer 70-300mm VR AF-S lens, do have the autofocus motor and thus will autofocus on the D5000.

That lens will work with the D3100, except for the autofocusing. In order to autofocus with the D3100, you need AF-S lenses, such as the AF-S VR Zoom-NIKKOR 70-300mm f/4.5-5.6G IF-ED or the
AF-S DX NIKKOR 55-300mm f/4.5-5.6G ED VR.

There is no such thing. You can see a full list of current Nikon lenses at
http://www.nikonusa.com/Find-Your-Nikon/Camera-Lenses/index.page

You can see a list of current and old Nikon autofocus lenses at
http://www.bythom.com/nikkordb.htm

You'll see that no 50mm Nikon lens has ever had VR.

I fix these all the time for this problem, The focus motor is bad, Tony

Nikon guarantees 5 year warranty to original purchaser. If your camera functions with another AF-S lens, then the problem lies with the 18-200mm lens. The AF motor unit needs replacement. Nikon should be able to repair the lens under warranty.
Make a copy of your receipt and go Nikon website to start repair service.

you should change the focus lock to AF-S, instead of AF-C

You might like to check if the A/M slider switch on the lens by the Nikon label is set to A.
It sometimes gets moved accidently.
Set to M the lens will not autofocus.

hi ! iam too a D40 holder,u can opt for AF-s versions or HSM sigma,both of them autofocus on d40,can go for 18-200 AFs VR or if u need still longer end go for 70-300 but it is costly than the sigms 70-300APO DG HSM MACRO..,its a much better reviewed lens and dont forget the cost..anyway do some research before considering..good luck

If it's an AF teleconverter you will be able to use it and get proper exposure from the camera, but it will not autofocus with the D60. The D60 requires AF-S lenses and teleconverters for autofocus.