nginx.conf

# Redirect every request to HTTPS...
server {
    listen 80;
    listen [::]:80;

    server_name .example.com;
    return 301 https://$host$request_uri;
}

# Redirect SSL to primary domain SSL...
server {
        listen 443 ssl http2;
    listen [::]:443 ssl http2;
    
    ssl_certificate /etc/nginx/ssl/example.com/449201/server.crt;
    ssl_certificate_key /etc/nginx/ssl/example.com/449201/server.key;

    ssl_protocols TLSv1.2;
    ssl_ciphers <REDACTED>;
    ssl_prefer_server_ciphers on;
    ssl_dhparam /etc/nginx/dhparams.pem;

    server_name www.example.com;
    return 301 https://example.com$request_uri;
}


server {
    listen 443 ssl http2;
    listen [::]:443 ssl http2;
    server_name example.com;
    root /home/forge/example.com;

    rewrite example.com.uk$ example.com/uk/ permanent;

    ssl_certificate /etc/nginx/ssl/example.com/449201/server.crt;
    ssl_certificate_key /etc/nginx/ssl/example.com/449201/server.key;

    ssl_protocols TLSv1.2;
    ssl_ciphers <REDACTED>;
    ssl_prefer_server_ciphers on;
    ssl_dhparam /etc/nginx/dhparams.pem;

    add_header X-Frame-Options "SAMEORIGIN";
    add_header X-XSS-Protection "1; mode=block";
    add_header X-Content-Type-Options "nosniff";

    index index.html index.htm index.php;

    charset utf-8;
    
    ## Wordpress single site settings

    location /site1/wp-json {
        rewrite ^/site1/wp-json(.*)$ /site1/?rest_route=$1;
    }

    location /site1/ {
       try_files $uri $uri/ /site1/index.php$is_args$args;
    }
    
    ## WordPress Multisite
    if (!-e $request_filename) {
        rewrite /wp-admin$ $scheme://$host$uri/ permanent;
        rewrite ^(/[^/]+)?(/wp-.*) $2 last;
        rewrite ^(/[^/]+)?(/.*\.php) $2 last;
    }
    set $cache_uri $request_uri;


    # POST requests and urls with a query string should always go to PHP
    if ($request_method = POST) {
        set $cache_uri 'NULL';
    }
    if ($query_string != "") {
        set $cache_uri 'NULL';
    }

    location /.well-known/acme-challenge {
        auth_basic off;
        allow all;
        alias /home/forge/.letsencrypt;
    }

    location  / {
        try_files $uri $uri/ /index.php?$query_string;
     }
    
    location ~ /\.(?!well-known).* {
        deny all;
    }

    location = /favicon.ico { access_log off; log_not_found off; }
    location = /robots.txt  { access_log off; log_not_found off; }

    access_log off;

    error_log  /var/log/nginx/example.com-error.log error;
    error_page 404 /index.php;
    
    location ~ \.php$ {
        fastcgi_split_path_info ^(.+\.php)(/.+)$;
        fastcgi_pass unix:/var/run/php/php7.0-fpm.sock;
        fastcgi_index index.php;
        include fastcgi_params;
    }

    location ~ /\.ht {
        deny all;
    }

    
        # BEGIN W3TC Minify cache
    location ~ /wp-content/cache/minify/.*\.js$ {
        types {}
        default_type application/x-javascript;
        expires modified 31536000s;
        add_header Vary "Accept-Encoding";
    }
    location ~ /wp-content/cache/minify/.*\.css$ {
        types {}
        default_type text/css;
        expires modified 31536000s;
        add_header Vary "Accept-Encoding";
    }
    location ~ /wp-content/cache/minify/.*js_gzip$ {
        gzip off;
        types {}
        default_type application/x-javascript;
        expires modified 31536000s;
        add_header Vary "Accept-Encoding";
        add_header Content-Encoding gzip;
    }
    location ~ /wp-content/cache/minify/.*css_gzip$ {
        gzip off;
        types {}
        default_type text/css;
        expires modified 31536000s;
        add_header Vary "Accept-Encoding";
        add_header Content-Encoding gzip;
    }
    # END W3TC Minify cache
    # BEGIN W3TC Minify core
    set $w3tc_enc "";
    if ($http_accept_encoding ~ gzip) {
        set $w3tc_enc _gzip;
    }
    if (-f $request_filename$w3tc_enc) {
        rewrite (.*) $1$w3tc_enc break;
    }
    rewrite ^/([a-z0-9]+/)?wp-content/cache/minify/ /index.php last;
    # END W3TC Minify core
    # BEGIN W3TC Page Cache core
    set $w3tc_rewrite 1;
    if ($request_method = POST) {
        set $w3tc_rewrite 0;
    }
    if ($query_string != "") {
        set $w3tc_rewrite 0;
    }
    if ($request_uri !~ \/$) {
        set $w3tc_rewrite 0;
    }
    if ($http_cookie ~* "(comment_author|wp\-postpass|w3tc_logged_out|wordpress_logged_in|wptouch_switch_toggle)") {
        set $w3tc_rewrite 0;
    }
    set $w3tc_preview "";
    if ($http_cookie ~* "(w3tc_preview)") {
        set $w3tc_preview _preview;
    }
    set $w3tc_ssl "";
    if ($scheme = https) {
        set $w3tc_ssl _ssl;
    }
    if ($http_x_forwarded_proto = 'https') {
        set $w3tc_ssl _ssl;
    }
    set $w3tc_enc "";
    if ($http_accept_encoding ~ gzip) {
        set $w3tc_enc _gzip;
    }
    if (!-f "$document_root/wp-content/cache/page_enhanced/$http_host/$request_uri/_index$w3tc_ssl$w3tc_preview.html$w3tc_enc") {
      set $w3tc_rewrite 0;
    }
    if ($w3tc_rewrite = 1) {
        rewrite .* "/wp-content/cache/page_enhanced/$http_host/$request_uri/_index$w3tc_ssl$w3tc_preview.html$w3tc_enc" last;
    }
    # END W3TC Page Cache core
    # BEGIN W3TC Page Cache cache
    location ~ /wp-content/cache/page_enhanced.*gzip$ {
        gzip off;
        types {}
        default_type text/html;
        add_header Content-Encoding gzip;
    }
    # END W3TC Page Cache cache
    # BEGIN W3TC Browser Cache
    gzip on;
    gzip_types text/css text/x-component application/x-javascript application/javascript text/javascript text/x-js text/richtext image/svg+xml text/plain text/xsd text/xsl text/xml image/bmp application/java application/msword application/vnd.ms-fontobject application/x-msdownload image/x-icon image/webp application/json application/vnd.ms-access application/vnd.ms-project application/x-font-otf application/vnd.ms-opentype application/vnd.oasis.opendocument.database application/vnd.oasis.opendocument.chart application/vnd.oasis.opendocument.formula application/vnd.oasis.opendocument.graphics application/vnd.oasis.opendocument.spreadsheet application/vnd.oasis.opendocument.text audio/ogg application/pdf application/vnd.ms-powerpoint application/x-shockwave-flash image/tiff application/x-font-ttf audio/wav application/vnd.ms-write application/font-woff application/font-woff2 application/vnd.ms-excel;
    location ~ \.(css|htc|less|js|js2|js3|js4)$ {
        expires 31536000s;
        etag on;
        if_modified_since exact;
        try_files $uri $uri/ $uri.html /index.php?$args;
    }
    location ~ \.(html|htm|rtf|rtx|svg|txt|xsd|xsl|xml)$ {
        etag on;
        if_modified_since exact;
        try_files $uri $uri/ $uri.html /index.php?$args;
    }
    location ~ \.(asf|asx|wax|wmv|wmx|avi|bmp|class|divx|doc|docx|eot|exe|gif|gz|gzip|ico|jpg|jpeg|jpe|webp|json|mdb|mid|midi|mov|qt|mp3|m4a|mp4|m4v|mpeg|mpg|mpe|mpp|otf|_otf|odb|odc|odf|odg|odp|ods|odt|ogg|pdf|png|pot|pps|ppt|pptx|ra|ram|svg|svgz|swf|tar|tif|tiff|ttf|ttc|_ttf|wav|wma|wri|woff|woff2|xla|xls|xlsx|xlt|xlw|zip)$ {
        expires 31536000s;
        etag on;
        if_modified_since exact;
        try_files $uri $uri/ $uri.html /index.php?$args;
    }
    location ~ \.(bmp|class|doc|docx|eot|exe|ico|webp|json|mdb|mpp|otf|_otf|odb|odc|odf|odg|odp|ods|odt|ogg|pdf|pot|pps|ppt|pptx|svg|svgz|swf|tif|tiff|ttf|ttc|_ttf|wav|wri|woff|woff2|xla|xls|xlsx|xlt|xlw)$ {
        etag off;
        if_modified_since off;
        try_files $uri $uri/ $uri.html /index.php?$args;
    }
    add_header Referrer-Policy "";
    # END W3TC Browser Cache
    
}

How about this?

    location /site1/ {
        rewrite ^/site1/wp-json(.*)$ /site1/?rest_route=$1;
       try_files $uri $uri/ /site1/index.php$is_args$args;
    }

and

    location /site1/ {
        rewrite ^/site1/wp-json(.*)$ /site1/?rest_route=$1 last;
       try_files $uri $uri/ /site1/index.php$is_args$args;
    }

@tarunlalwani, sadly neither of these fixed it