[time] Fix strftime %Z and %z returning GMT/+0000 instead of actual timezone

Since the timezone bloat elimination (#13635) removed setenv("TZ")/tzset()
on embedded platforms, libc's ::strftime no longer has access to timezone
state for %Z and %z format specifiers. The time values (hour, minute, etc.)
are correct since they come from our custom timezone parser via struct tm
fields, but %Z and %z read from libc's internal _tzname/_timezone which
are stuck at GMT/+0000.

Fix by substituting %Z and %z in the format string with the correct
"+HHMM"/"-HHMM" designation computed from our parsed timezone offset
before passing to ::strftime. The substitution only runs when %Z or %z
are present in the format string, so there is no overhead for the common
case.

Also refactors strftime_to() to delegate to strftime() instead of
duplicating the ::strftime call, ensuring both code paths get the fix.

esphome/components/time/posix_tz.cpp

@@ -4,6 +4,7 @@
 
 #include "posix_tz.h"
 #include <cctype>
+#include <cstdio>
 
 namespace esphome::time {
 
@@ -442,6 +443,18 @@ bool parse_posix_tz(const char *tz_string, ParsedTimezone &result) {
   return internal::parse_dst_rule(p, result.dst_end);
 }
 
+// Format a POSIX offset (positive = west) as "+HHMM" / "-HHMM" for display.
+// Convention: negate POSIX sign so east-of-UTC is positive (ISO 8601 / RFC 2822).
+void format_designation(int32_t posix_offset, char *buf, size_t buf_size) {
+  int32_t display = -posix_offset;
+  char sign = display >= 0 ? '+' : '-';
+  if (display < 0)
+    display = -display;
+  int h = display / 3600;
+  int m = (display % 3600) / 60;
+  snprintf(buf, buf_size, "%c%02d%02d", sign, h, m);
+}
+
 bool epoch_to_local_tm(time_t utc_epoch, const ParsedTimezone &tz, struct tm *out_tm) {
   if (!out_tm) {
     return false;

esphome/components/time/posix_tz.h

@@ -36,6 +36,9 @@ struct ParsedTimezone {
   bool has_dst() const { return this->dst_start.type != DSTRuleType::NONE; }
 };
 
+/// Format a POSIX offset as "+HHMM"/"-HHMM" into buf (must be >= 6 bytes).
+void format_designation(int32_t posix_offset, char *buf, size_t buf_size);
+
 /// Parse a POSIX TZ string into a ParsedTimezone struct.
 ///
 /// @deprecated Remove before 2026.9.0 (bridge code for backward compatibility).

esphome/core/time.cpp

@@ -2,6 +2,9 @@
 #include "helpers.h"
 
 #include <algorithm>
+#ifdef USE_TIME_TIMEZONE
+#include "esphome/components/time/posix_tz.h"
+#endif
 
 namespace esphome {
 
@@ -14,12 +17,45 @@ uint8_t days_in_month(uint8_t month, uint16_t year) {
 
 size_t ESPTime::strftime(char *buffer, size_t buffer_len, const char *format) {
   struct tm c_tm = this->to_c_tm();
+#ifdef USE_TIME_TIMEZONE
+  // ::strftime uses libc's internal timezone state for %Z and %z, but we
+  // eliminated setenv("TZ")/tzset() on embedded platforms to save flash.
+  // Substitute %Z and %z with correct values from our parsed timezone.
+  // Quick scan: does format contain %Z or %z?
+  bool needs_subst = false;
+  for (const char *p = format; *p; p++) {
+    if (*p == '%' && (*(p + 1) == 'Z' || *(p + 1) == 'z')) {
+      needs_subst = true;
+      break;
+    }
+  }
+  if (needs_subst) {
+    const auto &tz = time::get_global_tz();
+    char designation[6];  // "+HHMM" + null
+    int32_t offset = c_tm.tm_isdst > 0 ? tz.dst_offset_seconds : tz.std_offset_seconds;
+    time::format_designation(offset, designation, sizeof(designation));
+
+    char modified[STRFTIME_BUFFER_SIZE];
+    char *out = modified;
+    char *out_end = modified + sizeof(modified) - 1;
+    for (const char *p = format; *p && out < out_end; p++) {
+      if (*p == '%' && (*(p + 1) == 'Z' || *(p + 1) == 'z')) {
+        p++;  // skip the Z/z
+        for (const char *d = designation; *d && out < out_end; d++)
+          *out++ = *d;
+      } else {
+        *out++ = *p;
+      }
+    }
+    *out = '\0';
+    return ::strftime(buffer, buffer_len, modified, &c_tm);
+  }
+#endif
   return ::strftime(buffer, buffer_len, format, &c_tm);
 }
 
 size_t ESPTime::strftime_to(std::span<char, STRFTIME_BUFFER_SIZE> buffer, const char *format) {
-  struct tm c_tm = this->to_c_tm();
-  size_t len = ::strftime(buffer.data(), buffer.size(), format, &c_tm);
+  size_t len = this->strftime(buffer.data(), buffer.size(), format);
   if (len > 0) {
     return len;
   }