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Can anyone suggest some effective tools used for field to field and screen to screen testing of web apps where we can calculate the time taken to one field to another and one screen to another
Following are the performance testing requirements asked from a company from us 
The first two navigation measurements are client side, browser rendering and possibly javascript execution. You can use browser developer tools and javascript profiling toolbox to measure these interactions.
ref:
I don't know what your drop-down does but for reports and enquiries, there is server interaction involved now. There are two parts to this. The server call/response and the front end drawing and rendering. The draw/render part you can use the same tools as above to evaluate.
The server call/response will have to be tested separately, under concurrent call conditions. You will have to simulate a condition where several users are making calls to the server at the same time, and monitor how the server behaves and how quickly (or slowly) it is able to respond.
Some free/open source tools that enables this type of testing:
Some popular commercial tools:
I would take a look at the service New Relic offers. At first you want to verify that the application can actually deliver these performance requirements in a test setup, afterwards you want to monitor that you keep delivering under higher real user load on the production servers.
New Relic Browser is the only monitoring product that gives you everything you need—full visibility into the complete web page lifecycle for modern browser-side applications
See the full feature list at: http://newrelic.com/browser-monitoring/features
Its the only tool I know that monitors the web-application traffic from browser-to-browser taking into account even the rendering speed.
I think it injects a bit of JavaScript into the pages to monitor and report the full application stack end-2-end.
Possible alternatives (that I haven't used)
Some of your goals are unrealistic/unachievable. An example is the 10 millisecond field to field transition time. Most Monitors in desktop use refresh at the 60hz rate, which leaves each frame on the screen for 16-17ms. Some studies suggest that the barrier for human perceptibility is about 50ms. So, to even test this you would need specialized hardware and software that most of your users are not going to have deployed due to cost.
The remainder of the items are easily covered by a standard automated functional testing tool with built in timing capability. Even though automated functional tools have built in timers (well, most of the tools anyway), few people seem to take advantage of this capability for timing and thus collection of performance information earlier in the QA cycle, instead of leaving questions of performance to the "performance team."
Your last two items can be accomplished using a headless database query tool which returns the first byte of the returned data set. Some automated functional testing tools have the capability of connecting to a database directly to run such queries, but not all. Setting a timer is key here as well. You can also pull a profiling report to cross check the performance of a particular set of queries across a variety of parameter sets.
