Do pilots of commercial aircraft have ready access to a map or data showing nearest airports in case of a need to make an emergency landing?
Since you asked about commercial aircraft, I will answer for a modern airliner: The Boeing 777. The details will differ between aircraft models, but the general functionality should be available in every modern airliner.
With a modern airliner equipped with an EFIS (Electronic Flight Instrument System), there is no need to carry paper maps with possible diversion airports any more. Selecting a suitable alternate and navigating there can be done with the FMC (Flight Management Computer).
On the 777, the FMC has a page with alternate airports that are selected automatically from the navigation database, but can also be overridden by an uplink (where the airline dispatch sends suitable alternate airports to the aircraft) or manually (e.g. required for ETOPS flights with selected ETOPS alternates). This page looks something like this:

(Boeing 777 FCOMv2 11.43.21 - Flight Management, Navigation - FMC Descent and Approach)
The list of four airports is shown in 1 and the estimated time of arrival (ETA) as well as the fuel at the diversion airport is shown in 4 and 5 respectively. 2 allows a request of alternate airports from the airlines dispatch via ACARS. In order to divert to one of these airports, you select the respective line from 1 and the press DIVERT NOW (9). This will set up the flight plan with a direct line towards the selected airport.

Alternate Airport Diversions
ALTN page 1/2 data aids the flight crew in finding a suitable alternate airport.
The page displays up to four airports in ETA sequence. Each airport on the list has an XXXX ALTN page with more data. [...]
Three alternate airport uplinks can be received. ALTN LIST page 1/2 can get an uplink for the entire page or just the ALTN INHIB line. ALTN LIST page 2/2 can receive an uplink of alternate airport names.

(Boeing 777 FCOMv2 11.43.20 - Flight Management, Navigation - FMC Descent and Approach)
These four diversion airports can also be displayed on the Navigation Display (ND):

(Boeing 777 FCOMv2 10.10.35 - Flight Instruments, Displays - Controls and Indicators)
This is particularly helpful if adverse weather is in the area (e.g. thunderstorms) because the weather radar would depict areas with large amounts of precipitation on the ND as well. Should an alternate diversion airport be out of display range, it will be shown like this:


Displays the FMC or pilot selected primary alternate airport. Displays up to four alternate airports when the EFIS control panel ARPT map switch is selected on.
Offscale airports (those beyond 640NM with 640 scale selected), directional arrow relative to the airplane position or PLAN center point and distance (distance remove in PLAN).

(Boeing 777 FCOMv2 10.40.20 - Flight Instruments, Displays - Navigation Displays)
All of this allows a fast decision for an alternate diversion airport. Nonetheless, communication with ATC (Air Traffic Control) is of course essential in an emergency and ATC will assist in finding a suitable diversion airport if necessary. I don't quite agree with

I'm thinking that seconds could mean the difference between life and death [...].

If you only have seconds to make such a decision, it means you are at very low altitude, meaning either just after takeoff from your departure airport or already very close to your destination airport. In this case, the usual choice in case of an emergency is to return to the departure airport or continue to the destination. There are exceptions of course, like e.g. a takeoff alternate:

A take-off alternate aerodrome should be specified in the operational flight plan if either the meteorological conditions at the departure aerodrome are below the applicable landing minima or in case it should not be possible to return to the departure aerodrome for some other reason.

(SKYbrary - Alternate Aerodrome)
In this case the takeoff alternate would however be chosen before even departing, so no quick decision is necessary.