Updated to alpha1+1 - pretty much the same story for me unfortunately.
The following log is right after a disconnect when idle. I restarted the wifi module with M552 on device.

[[language]]
=== Diagnostics ===
Used output buffers: 3 of 32 (17 max)
=== Platform ===
RepRapFirmware for Duet WiFi version 1.20beta1 running on Duet WiFi 1.0
Board ID: 08DAM-9K9K2-NGNSN-6JKD0-3SJ6T-15UBM
Static ram used: 15128
Dynamic ram used: 96048
Recycled dynamic ram: 3512
Stack ram used: 1352 current, 4880 maximum
Never used ram: 11504
Last reset 00:06:35 ago, cause: reset button or watchdog
Last software reset reason: User, spinning module GCodes, available RAM 11504 bytes (slot 3)
Software reset code 0x0003, HFSR 0x00000000, CFSR 0x00000000, ICSR 0x00400000, BFAR 0xe000ed38, SP 0xffffffff
Error status: 0
Free file entries: 10
SD card 0 detected, interface speed: 20.0MBytes/sec
SD card longest block write time: 0.0ms
MCU temperature: min 41.1, current 41.6, max 42.8
Supply voltage: min 24.1, current 24.2, max 24.6, under voltage events: 0, over voltage events: 0
Driver 0: stalled standstill
Driver 1: stalled standstill
Driver 2: stalled standstill
Driver 3: standstill
Driver 4: standstill
Date/time: 2017-10-15 15:01:57
Slowest main loop (seconds): 0.008844; fastest: 0.000033
=== Move ===
MaxReps: 0, StepErrors: 0, FreeDm: 240, MinFreeDm 240, MaxWait: 0ms, Underruns: 0, 0
Scheduled moves: 0, completed moves: 0
Bed compensation in use: none
Bed probe heights: 0.000 0.000 0.000 0.000 0.000
=== Heat ===
Bed heater = 0, chamber heater = -1
Heater 0 is on, I-accum = 0.0
Heater 1 is on, I-accum = 0.3
=== GCodes ===
Segments left: 0
Stack records: 1 allocated, 0 in use
Movement lock held by null
http is idle in state(s) 0
telnet is idle in state(s) 0
file is idle in state(s) 0
serial is idle in state(s) 0
aux is idle in state(s) 0
daemon is idle in state(s) 0
queue is idle in state(s) 0
autopause is idle in state(s) 0
Code queue is empty.
Network state is running
WiFi module is connected to access point 
WiFi firmware version 1.20alpha1+1
WiFi MAC address 5c:cf:7f:ef:4c:24
WiFi Vcc 3.11, reset reason Turned on by main processor
WiFi flash size 4194304, free heap 35064
WiFi IP address 192.168.1.107
WiFi signal strength -58dBm
Reconnections 0
HTTP sessions: 1 of 8
Socket states: 2 0 0 0 0 0 0 0
Responder states: HTTP(1) HTTP(0) HTTP(0) HTTP(0) FTP(0) Telnet(0)

Feel bad piling on here, but same timeout issues in 1.19.2 for me. I've checked everything with M122. Slightly poor connection (-58db) but otherwise there is nothing outstanding I can find. My experience sounds very similar to Keegan's.

Some other details:
• Disconnects when not printing
• Always a timeout error
• Stays connected for about 30 seconds on average
• Uploading or sending commands often causes it to disconnect (CPU issue?)
• Cannot re-connect after (requires hard reset on board)

I was already getting a bunch of disconnects in 1.18 but I chalked that up to signal strength, but its gotten worse.
Let me know if there is any way I can help.