your topology is principally correct. It can be improved in various ways - how much improvement you can implement, that's down to your motivations in the project and your budget :-)
Your power supply should have an output voltage limit that does not overcharge the battery, and enough current capacity to keep your load alive at the lowest expected battery voltage level. What's left, flows into the battery when charging.
The PSU should not mind if something keeps its output down (possibly down to zero, if the battery ever gets over-discharged, which BTW is pretty destructive to LiIon-chemistry cells). Many power supplies just shut down to prevent damage, if they cannot sustain the nominal output voltage (within a relatively narrow margin). I.e., your power supply should have a "constant current limit" or should be nominally a battery charger (and a pretty stupid one at that) and you should consider the risk of fire, if the PSU keeps feeding a shorted circuitry (for whatever reason/defect).
In industrial practice of DC-supply systems with battery backup, you can see various solutions, starting from this one of yours on the cheap side. I'm speaking setups with 12 or 24 or 48 V nominal battery voltage (even higher levels are sometimes used: 110 or 220V DC). Conceptually, this topology with a step-down power supply working as a charger (often called a "rectifier" in the DC supplies) and the load connected (semi-)permanently to the battery is typical e.g. for telco switch power supplies at 48V DC (more like 55 V when fully charged). Solar systems look like this. Optionally you can have a step-up converter (also called an "inverter") on the battery-backed output, to power e.g. 230V AC mains devices - but if the actual loads need low-volt DC (12V nominal input and further buck converters inside), the intermediate up-conversion is often just a net loss of efficiency. I'm getting off topic here :-)
Things that can be improved about your circuit:
implement some fuse or a circuit breaker against overcurrent (short circuit). This is to prevent fire. Batteries can deliver astoundingly high short circuit currents
implement protection against battery over-discharge. This effectively means that you need a switch (or two) that detach the load and/or the battery. If you can build this protection logic out of micro-power devices (CMOS logic and MOSFET switches), it can be made to rely on the battery to "always have some juice left" to keep this electronics alive. That said, MOSFET's are not much use at 3.7V, because their V(gs) to fully open is typically more like 4 Volts. Note that the nominal Vgs(th) specifies where the FET "just starts to open a little", leaking some milliamps... This is probably one of the reasons why most "point of load" VRM's in PC motherboards typically use an input rail voltage of 12 Volts (and 19 Volts input in most notebooks, to be able to charge a battery to 14-ish Volts using a buck converter, on top of the 12V-ish internal mains rail).
a clever charger, built for "battery-backed DC UPS applications", that's trying to follow some charging curve towards the battery, should measure the load from the attached "consumption devices" and try to cater for them "separately", even in this principally "on line" topology. Or: measure the actual net charge current into the battery and try to keep a desired constant current at that measurement point (while not raising a battery failure alarm when the consuming devices "siphon" some extra current away). I.e., you'd need to enhance your circuit with several more regulation and protective feedbacks, "sensors" and "switches".
How much of this you implement is your own choice. Unless this is some portable or micropower device, consider using a higher battery voltage, say 12 Volts, if you want to improve that charger circuit and have some room for playing with MOSFET's. Also, there are off-the-shelf PSU modules (often DIN rail mount) for DC UPS applications, with internal logic and an extra pair of screw terminals for battery attachment.
Years ago, I've actually sketched and built a circuit that can be attached to the output of a strong enough and stupid "charger" PSU (13.5V and a constant current limit) to add battery over-discharge protection and to limit the charging current from the overly strong charger PSU. Designed for lead-acid batteries.
Looking back I have to say that the design was naive, and it would nowadays probably be stupid to reproduce that. Therefore I'm not including the source files (ask me if you desperately want them :-)
It is probably much easier nowadays to just use an off-the-shelf PSU with battery UPS function. There are some such modules by MeanWell: the DRC, SCP, AD and ADD series, with a varying set of protections - namely the SCP doesn't seem to have an over-discharge disconnector. And, I have recently discovered a neat family of stationary battery chargers (by a local vendor called BKE) that are designed to work as a DC UPS with an external battery, and even have a basic remote management capability over TCP/IP.