I don’t really see the UK dominating the game. That is generally the American’s role.
The enabler of US interference in Europe
The grinder against Italian expansion, and
The gatekeeper of Japanese expansion in the Pacific
With that being said, early in the game the UK wants to make Germany take as many losses as possible. In conjunction the UK wants to disable/weaken/delay any Italian control of the Med. Usually that just requires trading UK ships for Italian ships and then hopefully having remaining aircraft to prevent Italy from moving transports undefended to expand. If Italy only has 1 transport, she is not going to sacrifice it with UK aircraft capable of sinking it. Italy may not even move the TT without multiple ships to defend it - forcing Italy to stay in port for multiple turns.
Later in the game, once the threat to London has mostly passed, UK would be well served to build ships off of Canada that will be able to move off of Gib to support American transport rotations. Generally getting 1-2 carriers with the 4 Aircraft you plunked on London earlier in the game and a few DD screens is going to be enough to deter a German suicide run trying to sink transports. The US will have her own ships, preferably a few Cruisers with DD as screens to lie in wait to support landings in Europe.
Once the support is established off of Gib, and the US has moved some bigger ships in to replace your carriers, you can plop down 4-5 TT in the English Channel in one turn and dump the UK forces onto Normandy to reinforce the US units that survived. This may include UK aircraft if they won’t be exposed to being lost from an axis counter attack.
If you’ve gotten that going in Europe, unless Germany took Moscow with relative ease, Germany will end up losing as it cannot defend everywhere all at once.
For the Pacific, the UK is basically subject to the whims of Japanese aggression. Japan’s best play is to control the DEI and then convoy the UK out of the Pacific theater. Knowing that, forcing Japan to fight for each island is valuable in slowing the advance down. Doubly important is being in position to keep the Burma Road in China’s hands. Its more aggravating than anything because Japan does NOT want to funnel income into China, but Japan is obligated to do so or face losing income in the northern territories. This is why I prefer Mech early on for the UK Pacific, because it can be in India and also be 1 move from keeping the Burma road open.
UK in the Pacific rarely dominates unless Japan loses badly to the US or misplays itself out of position (which happens a lot). Just focus on being a thorn in Japan’s side early, defending Calcutta if Japan is full steam KIF and force Japan to take massive air unit losses so it cannot leverage simply replacing carriers and flying aircraft onto them.