I agree with @artem_d93 that BGP is the most appropriate solution. With BGP, when one path fails, re-convergence occurs and a new path (over the alternate ISP) is presented to network traffic. This can take several minutes to occur.
You really need to ask yourself or management how critical it is that these servers remain available if an ISP or a piece of network equipment fails. If it's not overkill and it's really necessary, one way to go might be to so something like I've done.
I have two Cisco 4431 routers for each ISP, each connected directly to the other, sharing a standby (HSRP) address. These connect to a pair of DMZ switches that are port channeled together (not a stack as a stack could fail). Then have two ASA 5515X that are connected so they share state and failover. Then on to two Nexus 5548UP switches at my network core. This is a fully redundant setup so if there is a failure anywhere along the line, we will keep running. Our IPv4 and IPv6 blocks are used for addresses between the ASA 5515x and ISR4431 routers. I also have a dedicated WAN router (that has other one off connections) connected back to my core as well as to the 4431 routers
Ask one of your ISPs to allocate you an IPv4 /24 block (the smallest possible). You will pay a monthly fee for this, but then it will be assigned for your needs as long as you stay with the ISP. You will need at least one business class router or routing firewall appliance that can handle BGP. Most of the dedicated routers can as well as some of the firewall appliances. Consumer grade equipment cannot.
Once you have a block assigned, you will need to get an ASN from RIPE. This Autonomous System Number will allow you to be a unique node on the internet.
Once you have redundant ISPs, you should have two routers that share an IP address (look up HSRP) in a standby/failover config, each router connected to one ISP. With HSRP, if one router fails, traffic will run over the other - the will share one of your assigned public addresses.
Between the previously mentioned routers and your private network, you should have a (or a pair) of firewall devices. And perhaps a pair of DMZ switches. With all of this interconnected, you can have a fully redundant internet connection.
Or you can have a pair of firewall devices that have all of this self contained. It can get pretty extensive, but if you really need it, it's really a solid way to go.
If you are accessing the addresses by IP address and port, give your end users both and tell them to use one if the other fails to load. It's clumsy work-around though.
If you have control over your DNS records, you can update the address, but make sure the TTL is set very low so stale records won't be around very long.
Use a web hosting service if possible - Some of them can provide VPN connectivity so the hosted servers can access internal data, however the type of access might make this unusable (such as database access).