Retina, 29 26 bytes

\\

<
10*
+`^(.*)¶
60*$1
.

Try it online! Uses newline separation, but link includes header to use spaces instead for convenience. Edit: Saved 3 bytes with help from @KevinCruijssen. Explanation:

\\

Delete the \s are they aren't necessary.

<
10*

Replace each < with 10 _s.

+`^(.*)¶
60*$1

Take the first line, multiply it by 60, and add the next line. Then repeat until there is only one line left.

.

Take the length i.e. convert to decimal.

Faster 51-byte version:

%`^(<*)(T*).*
$.(10*$1$2
+`^(.+)¶(.+)
$.($1*60*_$2*

Try it online! Uses newline separation, but link includes header to use spaces instead for convenience. Explanation:

%`^(<*)(T*).*
$.(10*$1$2

Match each line individually, and count the number of Ts and 10 times the number of <s. This converts each line into its base-60 "digit" value.

+`^(.+)¶(.+)
$.($1*60*_$2*

Base 60 conversion, running a line at a time. The computation is done in decimal for speed.

Retina, 29 26 23 bytes

<
10*T
+`^(.*)¶
60*$1
T

Try it online! Uses newline separation, but link includes header to use spaces instead for convenience. Edit: Saved 3 bytes with help from @KevinCruijssen. Saved a further 3 bytes thanks to @FryAmTheEggman. Explanation:

<
10*T

Replace each < with 10 Ts.

+`^(.*)¶
60*$1

Take the first line, multiply it by 60, and add the next line. Then repeat until there is only one line left.

T

Count the Ts.

Faster 51-byte version:

%`^(<*)(T*).*
$.(10*$1$2
+`^(.+)¶(.+)
$.($1*60*_$2*

Try it online! Uses newline separation, but link includes header to use spaces instead for convenience. Explanation:

%`^(<*)(T*).*
$.(10*$1$2

Match each line individually, and count the number of Ts and 10 times the number of <s. This converts each line into its base-60 "digit" value.

+`^(.+)¶(.+)
$.($1*60*_$2*

Base 60 conversion, running a line at a time. The computation is done in decimal for speed.

Retina, 29 bytes

\\

<+
10*$&
+`^(.*)¶
60*$1
.

Try it online! Uses newline separation, but link includes header to use spaces instead for convenience. Explanation:

\\

Delete the \s are they aren't necessary.

<+
10*$&

Multiply the <s by 10.

+`^(.*)¶
60*$1

Take the first line, multiply it by 60, and add the next line. Then repeat until there is only one line left.

.

Convert to decimal.

Faster 51-byte version:

%`^(<*)(T*).*
$.(10*$1$2
+`^(.+)¶(.+)
$.($1*60*_$2*

Try it online! Uses newline separation, but link includes header to use spaces instead for convenience. Explanation:

%`^(<*)(T*).*
$.(10*$1$2

Match each line individually, and count the number of Ts and 10 times the number of <s. This converts each line into its base-60 "digit" value.

+`^(.+)¶(.+)
$.($1*60*_$2*

Base 60 conversion, running a line at a time. The computation is done in decimal for speed.

Retina, 29 26 bytes

\\

<
10*
+`^(.*)¶
60*$1
.

Try it online! Uses newline separation, but link includes header to use spaces instead for convenience. Edit: Saved 3 bytes with help from @KevinCruijssen. Explanation:

\\

Delete the \s are they aren't necessary.

<
10*

Replace each < with 10 _s.

+`^(.*)¶
60*$1

Take the first line, multiply it by 60, and add the next line. Then repeat until there is only one line left.

.

Take the length i.e. convert to decimal.

Faster 51-byte version:

%`^(<*)(T*).*
$.(10*$1$2
+`^(.+)¶(.+)
$.($1*60*_$2*

Try it online! Uses newline separation, but link includes header to use spaces instead for convenience. Explanation:

%`^(<*)(T*).*
$.(10*$1$2

Match each line individually, and count the number of Ts and 10 times the number of <s. This converts each line into its base-60 "digit" value.

+`^(.+)¶(.+)
$.($1*60*_$2*

Base 60 conversion, running a line at a time. The computation is done in decimal for speed.