I have doubts that any of those powers were in any position to launch offensives in 1939 against Romania, let alone a joint Romanian-Polish alliance. Forgive me if Im coming off condescending, for that is not my intention. Maybe the U.S.S.R could try, but their performance in 1939 isn't exactly the greatest.
I thought Poland and Romania did have a defense agreement, but Poland, not wishing to alter their relationship with the Western Allies, did not initiate it when Germany invaded. If anything, I think Romania and Poland may have put up a good show against the old Whermact, but the threat of a Soviet intervention on the Romanian border may have kept any meaningful aide from reaching the Poles. But a demonstration of solidarity and unity may have stopped the Soviets from entering the conflict.
What can Romania bring to help the Polish really though? They'd need to properly mobilize and call up reserves first. From some quick Internet research it appears they had a decent fleet of Czech light tanks and some French R-35, their airforce is outdated though. In all as far as quality, Id say they are roughly on par with Poland in regards to equipment, not sure about how many divisions they had, but a few Romanian Army units may be enough to at least stabilize the front in Poland, give Poland time to finish mobilizing, organize counterattacks. Or they might suffer the same fate of being surrounded and cut off in pockets, as what befell the Poles. I think personally, it would make some difference. But superior tactics can often beat superior numbers.
You are correct, Poland DID have a defence agreement with Romania, and it was Poland who asked Romania to stay out.
If Poland and France ask Romania to join, I have little doubt it would - PM at the time was Armand Calinescu, who was very pro-British, and would see the move as a way to shore up his position.
However, this is where things get complicated.
What is the position of the other Balkan states and of the Soviet Union WRT this new situation? What do the British and French do? What actions does Romania take? It can all go in very different directions...
USSR
Stalin was a very, very cautios person, who never went into a fight he was unsure he could win. At the time, the Franco-British alliance looked very, very strong to the outside world, and Stalin had no reason to challenge it. Therefor, if Romanian, or even worse, British and French units, start arriving in south-eastern Poland (the area Polish planners called "the Romanian Bridgehead", where they intended to make their last stand), Stalin may very well postpone his invasion of Poland and wait for the Germans to occupy the region before pressing his claims; or, at the very least, only send in forces into the northern sector (a low-probability event IMO). In any case, he is almost certain to not open hostilities with Romania
Hungary
Hungary was no friend of Romania at the time. However, they were in absolutely no position to join the were, given the state of their army, and they knew it. They were still hampered by the the former restricitions imposed upon them by Trianon, having little modern equipment and an airforce that was still forming.
For example, during their invasion of Carpatho-Ukraine, they managed to muster
an infantry regiment, two cavalry regiments, three infantry battalions on bicycles, one motorized battalion, two border guard battalions, one artillery battalion and two armored trains.
During the war with Slovakia, 4 months prior, they managed to send
5 infantry battalions
2 cavalry battalions
1 motorised battalion
3 armoured cars
70 tankettes
5 light tanks
Given the fact that, by attacking Romania, Hungary would be declaring war one France and Britain as well, and the fact that there existed the ever-present danger that Yugoslavia would attack them as well, i have doubts Hungary's leadership would opt for such a risky move so early on. Instead, they would most likely sit on the sidelines and bide their time.
Bulgaria
They too had demands on Romania (southern Dobrogea, aka "the Cadrilater"), they too had a mediocre army, and they too were boxed in by potentially hostile states (Yugoslavia, Greece, Turkey), all of whom had good relations with Romania and prior defence agreements, and they too risked Anglo-French intervention. Therefor, I again doubt they would opt to go down such a risky path.
The Allies
Britain and France had quite a number of assets in the eastern Mediterranean, which could in theory have been transferred to Romania, either through the Turkish straights and unloaded in Constanta or via rail through Yugoslavia. If they get at least some token ground and air forces into southern Poland on time, it would greatly complicate Stalin's planning.
Assuming the above holds true, what we get is a greatly prolongued Polish campaign that sees several Polish divisions (most likely (1) the remnants of Operational group Boruta, itself the remnants of the Krakow army - 2 infantry divisions, 1 motorized brigade and 1 mountain brigade; (2) the remnants of Karpaty Army - 3 infantry divisions, 2 mountain brigades and assorted smaller units and (3) the remnants of Northern front, itself a remnant of Modlin army, totalling 39k men from a whole number of units) fall back to the south-east where they link up with several Romanian divisions and minor Franco-British forces. These probably force the over-extended panzer spearheads to halt and wait for the infantry to catch up, by which time the autumn rains should start.
With the roads turned to mud and the concentration of allied forces growing as they retreat towards an ever smaller chokepoint, it's unlikely the Germans, already very far away from their supply lines, manage an encirclement. Instead, the opposition probably continues to withdraw to the Romanian border.
Now the question we have to ask ourselves is whether or not France presses on with its Saar offensive, whether Hitler feels forced to withdraw units west if they do, and if so how many. Furthermore, we got to ask ourselves what strategic decision do the Germans make - do they leave Romania alone and strike France in May, or do they use the November-April period to invade Romania, or even forgo attacking France untill* Romania is completely conquered?. If they go for Romania though, the delay it entails and the attrition to their mechanized units and airforce almost surely means Sickle-Cut fails, if and when it's attempted.
* - given the winter weather, autumn and spring mud and the bad roads, this would mean a Romanian surrender sometime in May-June 1940, possibly as late as July