I suspect too high current pulses from the input, and the battery internal resistance causing issues.
I see some problems with the circuits ...
22 uH for inductors seem like very high value, and should also be adjusted according to the circuit needs. The 5v output doesn't need same inductor value as the one aiming for 25v, should optimize according to needs.
The datasheet for MT3608 says "The recommended values of inductor are 4.7 to 22μH. Small size and better efficiency are the major concerns for portable device, such as MT3608 used for mobile phone. The inductor should have low core loss at 1.2MHz and low DCR for better efficiency. To avoid inductor saturation current rating should be considered"
I'd probably test the circuit with a more reasonable inductor, like 6.8uH .. 10uH. 22uH seems a bit much, but then again you also don't say how much power/current on 5v , how much power/current on 25v
1n4148 is not particularly great, there's better options but it will work.
I don't see any input capacitance, the regulator needs it even when running from battery... I'd suggest 2 x ~ 10uF 25-50v rated ceramics in parallel, x5r , x7r at the input of each regulator ... 22uF ceramics may be more expensive.
On the output, for the 5v output 22uF ceramic would be fine, as long as voltage rating is up, 25v+ ... for the 25v out regulator, you'll want at least a 35v rated ceramic but it will practically be 1-2 uF in capacitance almost pointless at that output voltage. I'd suggest going with something like a 10uF 50v or higher rated ceramic in parallel with a 10-47uF 35v polymer capacitor.
You should look at the datasheets of other similar regulators... for example other 1.2 Mhz-ish step-up regulators. The datasheets for those may be more detailed and give you more info about what inductors are optimal , how to calculate things... of course due to slightly different internals of the chips, some of the info may not be
For example, I just googled for "1.2 mhz step-up regulator" and found LT1961 and MIC2619 and LT3426 ..
LT3426 :
https://www.analog.com/media/en/technical-documentation/data-sheets/3426fb.pdfMIC2619 :
http://ww1.microchip.com/downloads/en/DeviceDoc/mic2619.pdfLT1961 :
https://docs.rs-online.com/67d8/0900766b81412881.pdf (see the inductor choice section in datasheet)