Is the electrical setup of my (rental) house safe?
DANGER. You MUST switch off the main switch before handling fuses or MCBs on that type of fuseboard.
It's generally safe, but with some reservations.

The use of mixed MCBs and fuses is not in itself a problem. Wylex make retro-fit MCBs and bases to fit in place of the old fuse carriers and bases.

The retro-fit MCBs have a lower ability to clear high current faults (rupture capacity) than the fuses they replace, so may not be suitable in all installations and the electrician should have checked this.

The cover that goes over the fuses/MCBs must have an aperture cut to fit round the MCBs. Without the cover, the gap round the MCBs/fuses may not be finger-proof with the potential to touch live parts. Fuseboxes of this age usually have unguarded live busbars within. Also, the cover is required to contain any molten metal or arc that is emitted if a fuse blows. Without the cover, there is a risk of fire or arc burns.

The black service fuse terminations look okay, but I can't see the top edges which must not have any unblocked holes for cables.


However, the installation does not comply with the current (18th Edition, as Amended) Wiring Regulations, because:

no RCD on circuits
questionable ingress protection (touch safe) on fusebox at front with cover removed, on top side of fusebox where cables enter
non-metallic fusebox may not be combustion-resistant

In general, there is no requirement to upgrade an electrical installation, but there is in rental property.
The law (if you are in England) is the Electrical Safety Standards in the Private Rented Sector (England) Regulations 2020

The regulations came into force on 1 June 2020, they apply to new
tenancies from 1 July 2020 and existing tenancies from 1 April 2021.
The standards that should be met are set out in the 18th edition of
the Wiring Regulations.
The Regulations state that a landlord must ensure that electrical
safety standards are met, and that investigative or remedial work is
carried out if the report requires this.
The electrical installation should be safe for continued use. In
practice, if the report does not require investigative or remedial
work, the landlord will not be required to carry out any further work.
Landlords must obtain a report (usually an Electrical Installation Condition Report or EICR) from the person conducting the inspection
and test which explains its outcomes and any investigative or remedial
work required.
Landlords must then supply a copy of this report to the tenant within 28 days of the inspection and test, to a new tenant before they
occupy the premises, and to any prospective tenant within 28 days of
receiving a request for the report.

https://www.gov.uk/government/publications/electrical-safety-standards-in-the-private-rented-sector-guidance-for-landlords-tenants-and-local-authorities/guide-for-landlords-electrical-safety-standards-in-the-private-rented-sector
The lack of RCD on circuits would usually be coded C3 - Improvement required, but if there was also no supplementary bonding in the bathroom, or the sockets might be used for equipment outdoors, or there are other conditions applying, that warrants a C2 - Potentially dangerous and that will require remedial work that must be done.
https://www.electriciansforums.net/threads/eicr-and-rcds-is-it-a-c2-or-c3.172686/
and
https://professional-electrician.com/technical/napit-codebreakers-14/