The  $:/core/ui/Buttons/new-tiddler uses the  $:/core/ui/Actions/new-tiddler to create a new tiddler.
$:/core/ui/Actions/new-tiddler looks like this:
\define get-tags() $(textFieldTags)$ $(tagsFieldTags)$
\whitespace trim
<$vars textFieldTags={{$:/config/NewTiddler/Tags}} tagsFieldTags={{$:/config/NewTiddler/Tags!!tags}}>
<$action-sendmessage $message="tm-new-tiddler" tags=<<get-tags>>/>
</$vars>
Change the line with the <$action-sendmessage to
<$action-sendmessage $message="tm-new-tiddler" tags=<<get-tags>> source-url="https://"/>
The same thing also works with the new-image button. The action is:  $:/core/ui/Actions/new-image
Also see: Docs for the action-sendmessage
As a second option it would be possible to use the  ActionCreateTiddlerWidget in combination with the send-message and the WidgetMessage: tm-edit-tiddler to open the newly created tiddler.
ActionCreateTiddler has an advangage, if you want to use a template-tiddler instead of hardcoding the actions.