The phone knows and stores the most recent value for signal
strength from the serving cell (the one the phone's currently
camped on).

Likewise it also has Network Measurement Results (NMR).  This
includes signal strength, BSIC and BCCH values from the (up to)
6 strongest neighbouring cells.

The most recent negotiated Timing Advance (TA) value is also
stored.  This may not be the TA value from the serving cell (or
even one of the cells in the NMR list) if it came from a cell
having the same Location Area Code (LAC) as the serving cell.
But you can force the phone to interact with the network at any
time, and get a fresh TA value.  The TA number (0 to 63 range)
gives the distance to the serving cell in kilometres when
multiplied by 0.5535.

In general, access to this information is not easy.  I've read
and used this and more on a laptop serially connected to GSM
devices.  Many GSM modems have "AT" commands to enable all this
to be read.  Most handsets don't give you access to NMR or TA
in this way (although all but the cheapest Siemens phones
generally do).  I have not written any applications to run on a
phone itself.

John