Important notice, please consider, HTML5 client as browser page has no native access to your system unlike mstsc.exe that is native RDP client with direct system access. Also HTML5 client has to rely on values that your browser reports and if the browser provides wrong values then the only way to fix browsers values is to tell the program the correct time zone shifting as in example given below.

To adjust time zone for HTML5 client do following.

1. firstly go sure you have updated HTML5 client to the latest available version: AdminTool > License > Check for update

2. afterwards go sure that time zone redirection is enabled as in picture
2. if automatical redirection still displays wrong time then locate variable
    *\Clients\www\software\html5\settings.js > W.timeZoneOffsetShift = "0:0";
    and change it depending on your needs "-60:-1" or "60:1" according to the missed time shift reported by your browser.
    In example left part "-60:-1" would shift time 60 minutes forward/back, and right part would adjust time zone UTC+/- shift.
    And by "-60:0" you would adjust time only while time zone would remain same, or by "0:-1" time zone offset would differ.

Set own time zone name than default.
For the same time zone as example UTC -4 etc. there may exist many different names because many countries at same time are in same time zone, usually the displayed name has no disadvantages on the functionality if it matches the time line (in example UTC -4) but mismatches the default name, so the problem (if any) is only of cosmetical nature since it is technically impossible to recognize time zone name via browser.
Nonetheless to set own time zone name for automatical recognition from system regedit strings do following.

1. locate variable in *\Clients\www\software\html5\settings.js >
IMPORTANT: due changes in new HTML5 5.62 version the W.timeZonesRegistry was changed to W.timeZonesRegistryNew

W.timeZonesRegistryNew = "Dateline%UTC-11%Aleutian@Hawaiian%Marquesas%Alaskan@UTC-09%Pacific@UTC-08%Mountain@US Mountain%Central@Easter Island@Central America%Eastern@SA Pacific%Atlantic@Pacific SA@Venezuela%Newfoundland%Greenland@E. South America@Montevideo%UTC-02%Azores@Cape Verde%GMT@Greenwich%Romance@W. Central Africa%GTB@Kaliningrad%Russian%Iran%Caucasus%Afghanistan%West Asia%India%Nepal%Central Asia%Myanmar%Altai%China%North Korea%Aus Central W.%Korea%AUS Central@Cen. Australia@AUS Central%West Pacific@AUS Eastern@West Pacific%Lord Howe%Central Pacific%Russia Time Zone 11@New Zealand@Russia Time Zone 11%Chatham Islands%Tonga@Samoa@Tonga%Line Islands";

All names represent 39 different time zones
-12 > Dateline
-11 > UTC-11
-10 > Aleutian@Hawaiian
-9.5 > Marquesas
-9 > Alaskan@UTC-09
-8 > Pacific@UTC-08
-7 > Mountain@US Mountain
-6 > Central@Easter Island@Central America
-5 > Eastern@SA Pacific
-4 > Atlantic@Pacific SA@Venezuela
-3.5 > Newfoundland
-3 > Greenland@E. South America@Montevideo
-2 > UTC-02
-1 > Azores@Cape Verde
0 > GMT@Greenwich
1 > Romance@W. Central Africa
2 > GTB@Kaliningrad
3 > Russian
3.5 > Iran
4 > Caucasus
4.5 > Afghanistan
5 > West Asia
5.5 > India
5.75 > Nepal
6 > Central Asia
6.5 > Myanmar
7 > Altai
8 > China
8.5 > North Korea
8.75 >Aus Central W.
9 > Korea
9.5 > AUS Central@Cen. Australia@AUS Central
10 > West Pacific@AUS Eastern@West Pacific
10.5 > Lord Howe
11 > Central Pacific
12 > Russia Time Zone 11@New Zealand@Russia Time Zone 11
12.75 >Chatham Islands
13 > Tonga@Samoa@Tonga
14 > Line Islands

If displayed time zone in example is UTC -4 which stands for Atlantic by default (final displayed name is the choice of system) but wished is another name as example Central Brazilian then open in registry editor regedit.exe following key path
HKEY_LOCAL_MACHINE\SOFTWARE\Microsoft\Windows NT\CurrentVersion\Time Zones\ 
and look for the name of wished time zone path name and replace in variable settings.js > W.timeZonesRegistryNewmatching string Atlantic with new string Central Brazilian (the string Standard Time can be omitted) according to regedit path name.
Example   > W.timeZonesRegistryNew = "****%Venezuela%Atlantic%Newfoundland%*****";
change to > W.timeZonesRegistryNew = "****%Venezuela%Central Brazilian%Newfoundland%*****";
In new HTML5 5.62 version the format has changed, now the containing string may contain up to 3 different types of time zones.
As example Central@Easter Island@Central America consist of three time zone types, upper semisphere time transitionwinter-summer-winter like in USA, lower semisphere summer-winter-summer transition like in Brazil, and zone without winter/summer transition like in Russia.

Conclusion: finally time zone adjustment consist of three parts, displayed time in hours:minutes, displayed UTC+/- offset, where both values are adjustable by W.timeZoneOffsetShift, and the displayed name for the specific UTC+/- time zone adjustable by W.timeZonesRegistryNew.