There could be lots of issues. Since there's no specific product info given in your post about the model of splitter or the models of the TVs then all I can offer is some general advice.
The first area is power.
HDMI carries a 5V power rail but it is very low current (only 50mA which is nothing really). It's there to power the HDMI input on the receiving "sink" device (TV/AV Receiver) so that the handshaking required for HDCP works even if some of the sink device(s) are switched off.
In order for a splitter to work it needs power too. Cheap splitters often siphon off some power from the 5V rail. That's not such a big issue if the splitter is behind a TV and all the HDMI cables are short. But when power runs over distance then there's a losses in the cable become more significant. The losses also become larger at higher temperatures. In short, by the time the splitter has taken a bite out of the power, what's left might not be enough when the signal gets to the TV upstairs.
ANSWER:
1) If the splitter doesn't have a power socket and transformer then change it for one that does
Next, EDIDs.
Part of the handshake between HDMI devices is the EDID info. This is a basic "Hello, what are you and what are your capabilities" data exchange.
When you have multiple sink devices then you have two main issues. The first is differing resolutions on TVs. An unsophisticated source device will go with the lowest common denominator. So your fancy 3D capable 1080p TV will be hobbled to the abilities of the lower resolution non-3D TV because the source box can't work out that both can accept 1080i
The second issue is the EDID information itself. Some display and sink devices don't report their capabilities correctly. This is a much tougher (read: expensive) problem to solve. You should exhaust all other possibilities first before looking in to this.
Finally, the quality of the HDMI implementation on source and sync devices
Sky is renown for playing fast and loose with the HDMI specs. For a start, they underspecified their HDMI circuitry on early HD boxes. Ever notice how long it has taken them to get DD over HDMI? That's a feature that has been in HDMI from day one ten years ago, yet Sky only just added it in the last 18 months. There's also a catalogue of carnage caused by Sky messing up HDMI with its various firmware updates. Lots of manufacturers have found that their gear suddenly stops working with a Sky box source after a Sky f/w update.
It's rare now to get issues with displays, but not unheard of. More likely though is compatibility problems with intermediary devices such as AV amps, switches, splitters, matrix boxes, scalers etc. There's not really a lot you can do about that except try the current splitter without the AV Receiver in place, or maybe if your amp has pass-thru and you aren't using ARC then try the splitter on the output of the amp instead of after the Sky box. That might not be practical in your particular layout, but as general advice it still stands.
One other thought is a HDMI extender. Think of it as a booster but that goes at the display end. The Neet- HDMI Extender / Booster - Amplified ACTIVE Repeater - v1.3 is about £25 with the power supply.
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I do have other sources (WD Live for example) that I could use as a source to see if that makes a difference. From memory the HDMI splitter is powered but it was quite cheap. Will play again and see where I get to with these ideas. TVM.