Yes, it does look like a bug. I have gone in and updated several templates i.e. post.php, forum.php, topic.php and recent.php to address this. Ideally, it would be good to fix the wpforo_member_link() function in functions-template.php to include the following at the very start:

$member = esc_url(WPF()->member->profile_url($member));

Conclusion 1 - Logged in users are taken to the wpForo profile page when they click on their own link, while they are taken to the UM profile of other members. Is this meant to be the expected behaviour?

Very strange - can't say I noticed this before, but after trying to replicate your steps, it seems that this a bug.

This can be fixed easily (although I think it needs to be fixed in the official release as it is possibly a bug) by adding a line to the post.php through a theme override as follows. Find the following piece of code...

    foreach($posts as $key => $post) : ?>
        <?php $member = wpforo_member($post); $post_url = wpforo_post($post,'url'); ?>
        <div id="post-<?php echo intval($post) ?>" class="post-wrap wpfn-<?php echo ($key+1); ?><?php if( $post ) echo ' wpfp-first' ?>">

        <?php $member = esc_url(WPF()->member->profile_url($member)); ?>

    foreach($posts as $key => $post) : ?>
        <?php $member = wpforo_member($post); $post_url = wpforo_post($post,'url'); ?>
        <?php $member = esc_url(WPF()->member->profile_url($member)); ?>
        <div id="post-<?php echo intval($post) ?>" class="post-wrap wpfn-<?php echo ($key+1); ?><?php if( $post ) echo ' wpfp-first' ?>">

1) Installed the latest WordPress on my Flywheel Local server on Mac. WordPress version is 4.9.8.

2) Installed the latest version of Ultimate Member. Version 2.0.25.

3) Installed the latest version of wpForo. Version 1.5.3.

4) Created a test user (joe bloggs) and checked the UM profile for this user. No tab for forum was visible.

5) Enabled the option to replace forum profile with UM profile, created a topic on the forum using the Joe Bloggs account and clicked the through to the profile link within the topic on forum. I was taken to the forum profile page of Joe Bloggs i.e. NO UM integration seen. 

6) Created a test user called Jane Doe and logged in. Now clicked the profile link of Joe Bloggs within the topic and I was taken to the UM profile.

Conclusion 1 - Logged in users are taken to the wpForo profile page when they click on their own link, while they are taken to the UM profile of other members. Is this meant to be the expected behaviour?

7) Disable memory cache and I was able to recreate the issue of endless loop  that I reported earlier.

Conclusion 2 - The conflict between Memory Cache setting and Replace with UM Profile setting exists even on a completely fresh install of all the components.

Any thoughts?

The endless loop error occurs even for a fresh install like above. The error messages are:

Fatal error: Maximum function nesting level of '256' reached, aborting! in /app/public/wp-includes/l10n.php on line 840

Fatal error: Maximum function nesting level of '256' reached, aborting! in /app/public/wp-includes/wp-db.php on line 602

Within the detail, I see the same stack trace repeated again and again, which tells me it is getting into an endless loop somehow. 

I am facing a strange issue i.e. when I disable Memory Cache and Enable the Replace Forum Profile with Ultimate Member Profile, it appears to go into an endless loop on some of the core calls and crashes out. 

It does not crash if I enable the Memory Cache, however the integration with UM does not work i.e. I am still taken to the wpForo member profiles. The other integrations relating to Notifications and UM Tab are not working either.

I so want to use wpForo as it looks very promising, especially due to its UM integrations. @anonymous3542 you appear to be the defacto expert on this, so would really appreciate your help.